TOPICS SURROUNDING THE COMBINATORIAL ANABELIAN GEOMETRY OF HYPERBOLIC CURVES III: TRIPODS AND TEMPERED FUNDAMENTAL GROUPS YUICHIRO HOSHI AND SHINICHI MOCHIZUKI JUNE 2023 Abstract. Let Σ be a subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinal- ity one. In the present paper, we continue our study of the pro-Σ fundamental groups of hyperbolic curves and their associated con- figuration spaces over algebraically closed fields in which the primes of Σ are invertible. The focus of the present paper is on appli- cations of the theory developed in previous papers to the theory of tempered fundamental groups, in the style of André. These applications are motivated by the goal of surmounting two funda- mental technical difficulties that appear in previous work of André, namely: (a) the fact that the characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve that is given in earlier work of André is only proven for a quite limited class of hyperbolic curves, i.e., a class that is “far from generic”; (b) the proof given in earlier work of André of a certain key injectivity result, which is of central importance in establishing the theory of a “p-adic local analogue” of the well-known “global” theory of the Grothendieck-Teichmüller group, contains a fundamental gap. In the present paper, we surmount these technical difficulties by introduc- ing the notion of an “M-admissible”, or “metric-admissible”, outer automorphism of the profinite geometric fundamental group of a p-adic hyperbolic curve. Roughly speaking, M-admissible outer automorphisms are outer automorphisms that are compatible with the data constituted by the indices at the various nodes of the spe- cial fiber of the p-adic curve under consideration. By combining this notion with combinatorial anabelian results and techniques de- veloped in earlier papers by the authors, together with the theory of cyclotomic synchronization [also developed in earlier papers by the authors], we obtain a generalization of André’s characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve to the case of arbitrary hyperbolic curves [cf. (a)]. Moreover, by applying the theory of local contractibility of p-adic analytic spaces developed by Berkovich, we show that the techniques developed in the present and earlier papers by the authors allow one to relate the groups of M-admissible outer automorphisms treated in the present paper to the groups of outer automorphisms 2010 Mathematics Subject Classification. Primary 14H30; Secondary 14H10. Key words and phrases. anabelian geometry, combinatorial anabelian geometry, tempered fundamental group, tripod, Grothendieck-Teichmüller group, semi-graph of anabelioids, hyperbolic curve, configuration space. The first author was supported by Grant-in-Aid for Scientific Research (C), No. 24540016, Japan Society for the Promotion of Science. 1 2 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI of tempered fundamental groups of higher-dimensional configuration spaces [associated to the given p-adic hyperbolic curve]. These con- siderations allow one to “repair” the gap in André’s proof — albeit at the expense of working with M-admissible outer automorphisms — and hence to realize the goal of obtaining a “local analogue of the Grothendieck-Teichmüller group” [cf. (b)]. Contents Introduction 2 0. Notations and Conventions 11 1. Almost pro-Σ combinatorial anabelian geometry 12 2. Almost pro-Σ injectivity 26 3. Applications to the theory of tempered fundamental groups 54 References 103 Introduction Let Σ ⊆ Primes be a subset of the set of prime numbers Primes which is either equal to Primes or of cardinality one. In the present paper, we continue our study of the pro-Σ fundamental groups of hyper- bolic curves and their associated configuration spaces over algebraically closed fields in which the primes of Σ are invertible [cf. [MzTa], [CmbCsp], [NodNon], [CbTpI], [CbTpII]]. The focus of the present paper is on ap- plications of the theory developed in previous papers to the theory of tempered fundamental groups, in the style of [André]. Just as in previous papers, the main technical result that underlies our approach is a certain combinatorial anabelian result [cf. Theo- rem 1.11; Corollary 1.12], which may be summarized as a generaliza- tion of results obtained in earlier papers [cf., e.g., [NodNon], Theorem A; [CbTpII], Theorem 1.9] in the case of pro-Σ fundamental groups to the case of almost pro-Σ fundamental groups [i.e., maximal almost pro-Σ quotients of profinite fundamental groups — cf. Definition 1.1]. The technical details surrounding this generalization occupy the bulk of §1. In §2, we observe that the theory of §1 may be applied, via a similar argument to the argument applied in [NodNon] to derive [NodNon], Theorem B, from [NodNon], Theorem A, to obtain almost pro-Σ gen- eralizations [cf. Theorem 2.9; Corollary 2.10; Remark 2.10.1] of the injectivity portion of the theory of combinatorial cuspidalization [i.e., [NodNon], Theorem B]. In the final portion of §2, we discuss the theory of almost pro-l commensurators of tripods [i.e., copies of the [geomet- ric fundamental group of the] projective line minus three points — cf. Lemma 2.12, Corollary 2.13], in the context of the theory of the tripod homomorphism developed in [CbTpII], §3. Just as in the case of the COMBINATORIAL ANABELIAN TOPICS III 3 theory of §1, the theory of §2 is conceptually not very difficult, but technically quite involved. Before proceeding, we recall that a substantial portion of the theory of [André] revolves around the study of outomorphism [i.e., outer auto- morphism] groups of the tempered geometric fundamental group of a p-adic hyperbolic curve, from the point of view of the goal of es- tablishing a p-adic local analogue of the well-known theory of the Grothendieck-Teichmüller group [i.e., which appears in the context of hyperbolic curves over num- ber fields]. From the point of view of the theory of the present series of papers, out- omorphisms of such tempered fundamental groups may be thought of as [i.e., are equivalent to — cf. Remark 3.3.1; Proposition 3.6, (iii); Re- mark 3.13.1, (i)] outomorphisms of the profinite geometric fundamen- tal group that are “G-admissible” [cf. Definition 3.7, (i)], i.e., preserve the graph-theoretic structure on the profinite geometric fundamental group. In a word, the essential thrust of the applications to the theory of tempered fundamental groups given in the present paper may be summarized as follows: By replacing, in effect, the G-admissible outomorphism groups that [modulo the “translation” discussed above] appear throughout the theory of [André] by “M-ad- missible” outomorphism groups — i.e., groups of out- omorphisms of the profinite geometric fundamental group that preserve not only the graph-theoretic struc- ture on the profinite geometric fundamental group, but also the [somewhat finer] metric structure on the var- ious dual graphs that appear [i.e., the various indices at the nodes of the special fiber of the p-adic curve under consideration — cf. Definition 3.7, (ii)] — it is possible to overcome various significant technical difficulties that appear in the theory of [André]. Here, we recall that the two main technical difficulties that appear in the theory of [André] may be described as follows: • The characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve that is given in [André], Theorems 7.2.1, 7.2.3, is only proven for a quite limited class of hyperbolic curves [i.e., a class that is “far from generic” — cf. [MzTa], Corollary 5.7], which are “closely related to tripods”. • The proof given in [André] of a certain key injectivity result, which is of central importance in establishing the theory of a 4 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI “local analogue of the Grothendieck-Teichmüller group”, con- tains a fundamental gap [cf. Remark 3.19.1]. In the present paper, our approach to surmounting the first technical difficulty consists of the following result [cf. Theorems 3.17, (iv); 3.18, (i)], which asserts, roughly speaking, that the theory of the tripod homomorphism developed in [CbTpII], §3, is compatible with the property of M-admissibility. Theorem A (Metric-admissible outomorphisms and the tripod homomorphism). Let n ≥ 3 be an integer; (g, r) a pair of nonnega- tive integers such that 2g − 2 + r > 0; p a prime number; Σ a set of prime numbers such that Σ  = {p}, and, moreover, is either equal to the set of all prime numbers or of cardinality one; R a mixed character- istic complete discrete valuation ring of residue characteristic p whose residue field is separably closed; K the field of fractions of R; K an algebraic closure of K; log X K a smooth log curve of type (g, r) over K. Write (X K ) log n for the n-th log configuration space [cf. the discussion entitled def log log over K; (X K ) log “Curves” in [CbTpII], §0] of X K n = (X K ) n × K K; def Σ Π n = π 1 ((X K ) log n ) for the maximal pro-Σ quotient of the log fundamental group of tpd be a 1-central {1, 2, 3}-tripod of Π n [cf. [CbTpII], (X K ) log n . Let Π Definitions 3.3, (i); 3.7, (ii)]. Then the restriction of the tripod ho- momorphism associated to Π n T Π tpd : Out FC (Π n ) −→ Out C (Π tpd ) [cf. [CbTpII], Definition 3.19] to the subgroup Out FC (Π n ) M ⊆ Out FC (Π n ) of M-admissible outomorphisms [cf. Definition 3.7, (iii)] factors through the subgroup Out(Π tpd ) M ⊆ Out C (Π tpd ) [cf. Definition 3.7, (i), (ii); Remark 3.13.1, (i), (ii)], i.e., we have a natural commuta- tive diagram of profinite groups Out FC (Π n ) M −−−→ Out(Π tpd ) M ⏐ ⏐ ⏐ ⏐   Out FC (Π n ) −−−→ Out C (Π tpd ) . T Πtpd Theorem A has the following formal consequence, namely, a gener- alization of the characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve that is given in [André], Theorems 7.2.1, 7.2.3, to arbitrary hyperbolic curves, albeit at COMBINATORIAL ANABELIAN TOPICS III 5 the expense of, in effect, replacing “G-admissibility” by the stronger condition of “M-admissibility” [cf. Corollary 3.20; Remark 3.20.1]. This generalization may also be regarded as a sort of strong version of the Galois injectivity result given in [NodNon], Theorem C [cf. Re- mark 3.20.2]. Theorem B (Characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve). Let F be a number field, i.e., a finite extension of the field of rational numbers; p a nonarchimedean prime of F ; F p an algebraic closure of the p-adic completion F p of F ; F ⊆ F p the algebraic closure of F in ∧ F p ; X F log a smooth log curve over F . Write F p for the completion def def def of F p ; G p = Gal(F p /F p ) ⊆ G F = Gal(F /F ); X F log = X F log × F F ; π 1 (X F log ) for the log fundamental group of X F log [which, in the following, we ∧ identify with the log fundamental groups of X F log × F F p , X F log × F F p — cf. the definition of F !]; ∧ π 1 temp (X F log × F F p ) ∧ for the tempered fundamental group of X F log × F F p [cf. [André], §4]; ρ X log : G F −→ Out(π 1 (X F log )) F for the natural outer Galois action associated to X F log ; ∧ ρ temp : G p −→ Out(π 1 temp (X F log × F F p )) X log ,p F for the natural outer Galois action associated to X F log × F F p [cf. [André], Proposition 5.1.1]; ∧ Out(π 1 (X F log )) M ⊆ ( Out(π 1 temp (X F log × F F p )) ⊆ ) Out(π 1 (X F log )) for the subgroup of M-admissible outomorphisms of π 1 (X F log ) [cf. Def- inition 3.7, (i), (ii); Proposition 3.6, (i)]. Then the following hold: factors through the subgroup (i) The outer Galois action ρ temp X log ,p F ∧ Out(π 1 (X F log )) M ⊆ Out(π 1 temp (X F log × F F p )). (ii) We have a natural commutative diagram G p −−−→ Out(π 1 (X F log )) M ⏐ ⏐ ⏐ ⏐   ρ X log F G F −−− → Out(π 1 (X F log )) 6 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI — where the vertical arrows are the natural inclusions, the upper horizontal arrow is the homomorphism arising from the factorization of (i), and all arrows are injective. (iii) The diagram of (ii) is cartesian, i.e., if we regard the various groups involved as subgroups of Out(π 1 (X F log )), then we have an equality G p = G F ∩ Out(π 1 (X F log )) M . One central technical aspect of the theory of the present paper lies in the equivalence [cf. Theorem 3.9] between the M-admissibility of outomorphisms of the profinite geometric fundamental group of the given p-adic hyperbolic curve and the I-admissibility [i.e., roughly speaking, compatibility with the outer action, by some open subgroup of the inertia group of the absolute Galois group of the base field, on an arbitrary almost pro-l quotient of the profinite geometric fun- damental group — cf. Definition 3.8] of such outomorphisms. This equivalence is obtained by applying the theory of cyclotomic syn- chronization developed in [CbTpI], §5. Once this equivalence is es- tablished, the almost pro-l injectivity results obtained in §2 then al- low us to conclude that this M-admissibility of outomorphisms of the profinite geometric fundamental group of the given p-adic hyperbolic curve is, in fact, equivalent to the I-admissibility of any [necessarily unique!] lifting of such an outomorphism to an outomorphism of the profinite geometric fundamental group of a higher-dimensional con- figuration space associated to the given p-adic hyperbolic curve [cf. Theorem 3.17, (ii)]. Finally, by combining this “higher-dimensional I-admissibility” with the combinatorial anabelian theory of [CbTpII], §1, we conclude [cf. Proposition 3.16, (i); Theorem 3.17, (ii)] that a certain “higher-dimensional G-admissibility” also holds, i.e., that the lifted outomorphism of the profinite geometric fundamental group of a higher-dimensional configuration space associated to the given p- adic hyperbolic curve preserves the graph-theoretic structure not only on the profinite geometric fundamental group of the original hyperbolic curve, but also on the profinite geometric fundamental groups of the various successive fibers of the higher-dimensional configuration space under consideration. In a word, it is precisely by applying this chain of equivalences — which allows us to control the graph-theoretic struc- ture of the successive fibers of the higher-dimensional configuration space under consideration — that allow us to surmount the two main technical difficulties dis- cussed above that appear in the theory of [André]. Put another way, if, instead of considering M-admissible outomor- phisms [i.e., of the profinite geometric fundamental group of the given COMBINATORIAL ANABELIAN TOPICS III 7 p-adic hyperbolic curve], one considers arbitrary G-admissible outo- morphisms [of the profinite geometric fundamental group of the given p-adic hyperbolic curve, as is done, in effect, in [André]], then there does not appear to exist, at least at the time of writing, any effective way to control the graph-theoretic structure on the successive fibers of higher-dimensional configuration spaces. In this context, we recall that in the theory of [CbTpII], a result is obtained concerning the preservation of the graph-theoretic structure on the successive fibers of higher-dimensional configuration spaces [cf. [CbTpII], Theorem 4.7], in the context of pro-l geometric fundamental groups. The significance, however, of the theory of the present paper is that it may be applied to almost pro-l geometric fundamental groups, i.e., where the order of the finite quotient implicit in the term “almost” is allowed to be divisible by p. Once one establishes the “higher-dimensional G-admissibility” dis- cussed above, it is then possible to apply the theory of local contractibil- ity of p-adic analytic spaces developed in [Brk] to construct from the given outomorphism of a profinite geometric fundamental group [of a higher-dimensional configuration space] an outomorphism of the corre- sponding tempered fundamental group [cf. Proposition 3.16, (ii)]. This portion of the theory may be summarized as follows [cf. Theorem 3.19, (ii)]. Theorem C (Metric-admissible outomorphisms and tempered fundamental groups). Let n be a positive integer; (g, r) a pair of nonnegative integers such that 2g − 2 + r > 0; p a prime number; Σ a nonempty set of prime numbers such that Σ  = {p}, and, moreover, if n ≥ 2, then Σ is either equal to the set of all prime numbers or of cardinality one; R a mixed characteristic complete discrete valuation ring of residue characteristic p whose residue field is separably closed; K the field of fractions of R; K an algebraic closure of K; log X K a smooth log curve of type (g, r) over K. Write (X K ) log n for the n-th log configuration space [cf. the discussion entitled def log log “Curves” in [CbTpII], §0] of X K over K; (X K ) log n = (X K ) n × K K; def Σ Π n = π 1 ((X K ) log n ) for the maximal pro-Σ quotient of the log fundamental group of ∧ (X K ) log n ; K for the p-adic completion of K; ∧ π 1 temp ((X K ) log n × K K ) for the tempered fundamental group [cf. [André], §4, as well as the discussion of Definition 3.1, (ii), of the present paper] of (X K ) log n × K 8 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∧ K ; ∧ def Π tp = lim π 1 temp ((X K ) log n n × K K )/N ←− N ∧ for the Σ-tempered fundamental group of (X K ) log [cf. n × K K [CmbGC], Corollary 2.10, (iii)], i.e., the inverse limit given by allow- ∧ ing N to vary over the open normal subgroups of π 1 temp ((X K ) log n × K K ) such that the quotient by N corresponds to a topological covering [cf. [André], §4.2, as well as the discussion of Definition 3.1, (ii), of the present paper] of some finite log étale Galois covering of ∧ (X K ) log of degree a product of primes ∈ Σ. [Here, we recall n × K K that, when n = 1, such a “topological covering” corresponds to a “com- binatorial covering”, i.e., a covering determined by a covering of the dual semi-graph of the special fiber of the stable model of some finite ∧ log étale covering of (X K ) log n × K K .] Write M tp Out FC (Π tp n ) ⊆ Out(Π n ) for the inverse image of Out FC (Π n ) M ⊆ Out(Π n ) [cf. Definition 3.7, (iii)] via the natural homomorphism Out(Π tp n ) → Out(Π n ) [cf. Propo- sition 3.3, (i)]. Then the resulting natural homomorphism FC M M Out FC (Π tp n ) −→ Out (Π n ) is split surjective, i.e., there exists a homomorphism M Φ : Out FC (Π n ) M −→ Out FC (Π tp n ) such that the composite Φ FC M M Out FC (Π n ) M −→ Out FC (Π tp n ) −→ Out (Π n ) is the identity automorphism of Out FC (Π n ) M . Up till now, in the present discussion, the p-adic hyperbolic curve un- der consideration was arbitrary. If, however, one specializes the theory discussed above to the case of tripods [i.e., copies of the projective line minus three points], then one obtains the desired p-adic local analogue of the theory of the Grothendieck-Teichmüller group, by considering the “metrized Grothendieck-Teichmüller group GT M ” as follows [cf. Theorem 3.17, (iv); Theorem 3.18, (ii); Theorem 3.19, (ii); Remarks 3.19.2, 3.20.3]. Theorem D (Metric-admissible outomorphisms and tripods). In the notation of Theorem C, suppose that (g, r) = (0, 3). Write Out F (Π n ) Δ+ ⊆ Out F (Π n ) for the inverse image via the natural homomorphism Out F (Π n ) → Out(Π 1 ) [cf. [CbTpI], Theorem A, (i)] of Out C (Π 1 ) Δ+ ⊆ Out(Π 1 ) COMBINATORIAL ANABELIAN TOPICS III 9 [cf. [CbTpII], Definition 3.4, (i)]; def Out FC (Π n ) Δ+ = Out F (Π n ) Δ+ ∩ Out FC (Π n ) [cf. Remark 3.18.1]; def Out F (Π n ) MΔ+ = Out F (Π n ) Δ+ ∩ Out F (Π n ) M ; def Out FC (Π n ) MΔ+ = Out FC (Π n ) Δ+ ∩ Out F (Π n ) M . Then the following hold: (i) We have equalities Out F (Π n ) Δ+ = Out FC (Π n ) Δ+ , Out F (Π n ) MΔ+ = Out FC (Π n ) MΔ+ . Moreover, the natural homomorphisms of profinite groups Out FC (Π n+1 ) Δ+ −−−→ Out FC (Π n ) Δ+       Out F (Π n+1 ) Δ+ −−−→ Out F (Π n ) Δ+ Out FC (Π n+1 ) MΔ+ −−−→ Out FC (Π n ) MΔ+       Out F (Π n+1 ) MΔ+ −−−→ Out F (Π n ) MΔ+ are bijective for n ≥ 1. In the following, we shall identify the various groups that occur for varying n by means of these natural isomorphisms and write def GT M = Out F (Π n ) MΔ+ = Out FC (Π n ) MΔ+ def ⊆ GT = Out F (Π n ) Δ+ = Out FC (Π n ) Δ+ [cf. [CmbCsp], Remark 1.11.1]. (ii) Write MΔ+ ⊆ Out(Π tp Out FC (Π tp n ) n ) for the inverse image of GT M ⊆ Out(Π n ) [cf. (i)] via the natu- ral homomorphism Out(Π tp n ) → Out(Π n ) [cf. Proposition 3.3, (i)]. Then the resulting natural homomorphism MΔ+ −→ GT M Out FC (Π tp n ) is split surjective, i.e., there exists a homomorphism MΔ+ Φ GT : GT M −→ Out FC (Π tp n ) such that the composite Φ GT MΔ+ GT M −→ Out FC (Π tp −→ GT M n ) 10 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI is the identity automorphism of GT M . In closing, we recall that “conventional research” concerning the Grothendieck-Teichmüller group GT tends to focus on the issue of whether or not the natural inclusion of the absolute Galois group of Q G Q → GT is, in fact, an isomorphism [cf. the discussion of [CbTpII], Remark 3.19.1]. By contrast, one important theme of the present series of papers lies in the point of view that, instead of pursuing the issue of whether or not GT is literally isomorphic to G Q , it is perhaps more natural to concentrate on the issue of verifying that GT exhibits analogous behavior/properties to G Q [or Q]. From this point of view, the theory of tripod synchronization and surjectivity of the tripod homomorphism developed in [CbTpII] [cf. [CbTpII], Theorem C, (iii), (iv), as well as the following discus- sion] may be regarded as an abstract combinatorial analogue of the scheme-theoretic fact that Spec Q lies under all characteristic zero schemes/algebraic stacks in a unique fashion — i.e., put another way, that all morphisms between schemes and moduli stacks that occur in the theory of hyperbolic curves in characteristic zero are compatible with the respective structure morphisms to Spec Q. In a similar vein, the theory of the subgroup GT M ⊆ GT developed in the present paper may be regarded as an abstract combinatorial analogue of the various decomposition subgroups G p ⊆ G F (⊆ G Q ) [cf. Theorem B] as- sociated to nonarchimedean primes. In particular, from the point of view of pursuing “abstract behavioral similarities” to the subgroups G p ⊆ G F (⊆ G Q ), it is natural to pose the question: Is the subgroup GT M ⊆ GT commensurably terminal? Unfortunately, in the present paper, we are only able to give a partial answer to this question. That is to say, we show [cf. Theorem 3.17, (v), and its proof; Remark 3.20.1] the following result. [Here, we remark that although this result is not stated explicitly in Theorem 3.17, (v), it follows by applying to GT M the argument, involving l-graphically full actions, that was applied, in the proof of Theorem 3.17, (v), to “Out FC (Π n ) M ”.] Theorem E (Commensurator of the metrized Grothendieck- -Teichmüller group). In the notation of Theorem D [cf., especially, the bijections of Theorem D, (i)], the commensurator of GT M in Out F (Π n ) is contained in the subgroup Out G (Π n ) ⊆ Out FC (Π n ) COMBINATORIAL ANABELIAN TOPICS III 11 of outomorphisms that satisfy the condition of “higher-dimensional G-admissibility” discussed above [cf. Definition 3.13, (iv); Remark 3.13.1, (ii)]. In particular, the commensurator of GT M in GT is contained in     G def G FC Out (Π n ) ⊆ Out (Π n ) ⊆ Out(Π 1 ) GT = GT ∩ n≥1 n≥1 [cf. the injections Out FC (Π n+1 ) → Out FC (Π n ) of [NodNon], Theorem B]. Acknowledgment The authors would like to thank E. Lepage for helpful discussions concerning the theory of Berkovich spaces and Y. Iijima for informing us of [Prs]. 0. Notations and Conventions Topological groups: Let G be a profinite group and Σ a nonempty set of prime numbers. Then we shall write G Σ for the maximal pro-Σ quotient of G. Let G be a profinite group and G  Q, Q  quotients of G. Then we shall say that the quotient Q dominates the quotient Q  if the natural surjection G  Q  factors through the natural surjection G  Q. 12 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI 1. Almost pro-Σ combinatorial anabelian geometry In the present §1, we discuss almost pro-Σ analogues of results on combinatorial anabelian geometry developed in earlier papers of the authors. In particular, we obtain almost pro-Σ analogues of combina- torial versions of the Grothendieck Conjecture for outer representations of NN- and IPSC-type [cf. Theorem 1.11; Corollary 1.12 below]. These almost pro-Σ analogues of combinatorial versions of the Grothendieck Conjecture will be applied in §2, together with various standard tech- niques of combinatorial anabelian geometry, to obtain almost pro-Σ analogues of certain standard injectivity results that will be applied in §3 to derive fundamental results concerning the outer representations of Galois groups that arise from hyperbolic curves and their associated configuration spaces over p-adic fields. In the present §1, let Σ ⊆ Σ † be nonempty sets of prime numbers and G a semi-graph of anabelioids of pro-Σ † PSC-type. Write G for the underlying semi-graph of G, Π G for the [pro-Σ † ] fundamental group of G, and G  → G for the universal covering of G corresponding to Π G . Definition 1.1. Let G be a profinite group, N ⊆ G a normal open subgroup of G, and G  Q a quotient of G. Then we shall say that Q is the maximal almost pro-Σ quotient of G with respect to N if the kernel of the surjection G  Q is the kernel of N  N Σ [cf. the discussion entitled “Topological groups” in §0], i.e., Q = G/Ker(N  N Σ ). Thus, Q fits into an exact sequence of profinite groups 1 −→ N Σ −→ Q −→ G/N −→ 1 . [Note that since N is normal in G, and the kernel Ker(N  N Σ ) of the natural surjection N  N Σ is characteristic in N , it holds that Ker(N  N Σ ) is normal in G.] We shall say that Q is a maximal almost pro-Σ quotient of G if Q is the maximal almost pro-Σ quotient of G with respect to some normal open subgroup of G. Lemma 1.2 (Properties of maximal almost pro-Σ quotients). Let G be a profinite group. Then the following hold. (i) Let N ⊆ G be a normal open subgroup of G and G  J a quotient of G. Write N J ⊆ J for the image of N in J. [Thus, N J is a normal open subgroup of J.] Then the quotient of J determined by the maximal almost pro-Σ quotient [cf. Defini- tion 1.1] of G with respect to N , i.e., the quotient of J by the image of Ker(N  N Σ ) in J, is the maximal almost pro-Σ quotient of J with respect to N J . (ii) Let N ⊆ G be a normal open subgroup of G and H ⊆ G a closed subgroup of G. If the natural homomorphism (N ∩ H) Σ → N Σ COMBINATORIAL ANABELIAN TOPICS III 13 is injective, then the image of H in the maximal almost pro- Σ quotient of G with respect to N is the maximal almost pro-Σ quotient of H with respect to N ∩ H. (iii) Let H ⊆ G be a normal closed subgroup of G and H  H ∗ a maximal almost pro-Σ quotient of H. Suppose that H is topo- logically finitely generated. Then there exists a maximal almost pro-Σ quotient H  H ∗∗ of H which dominates H  H ∗ [cf. the discussion entitled “Topological groups” in §0] such that the kernel of H  H ∗∗ is normal in G. Proof. Assertions (i), (ii) follow immediately from the various defini- tions involved. Next, we verify assertion (iii). Let N ⊆ H be a normal open subgroup of H with respect to which H ∗ is the maximal almost pro-Σ quotient of H. Now since H is topologically finitely generated, and N ⊆ H is open, it follows that there exists a characteristic open subgroup J ⊆ H such that J ⊆ N . Observe that since H is normal in G, and J is characteristic in H, it holds that J is normal in G. Thus, if we write H ∗∗ for the maximal almost pro-Σ quotient of H with respect to J, then H ∗∗ satisfies the conditions of assertion (iii). This completes the proof of assertion (iii).  Definition 1.3. Let I be a profinite group and ρ : I → Aut(G) ⊆ Out(Π G ) a continuous homomorphism. Then we shall say that ρ is of PIPSC-type [where the “PIPSC” stands for “potentially IPSC”] if the following conditions are satisfied: † (i) I is isomorphic to Z Σ as an abstract profinite group. (ii) there exists an open subgroup J ⊆ I such that the restriction of ρ to J is of IPSC-type [cf. [NodNon], Definition 2.4, (i)]. Lemma 1.4 (Profinite Dehn multi-twists and finite étale cov- erings). Let α ∈ Out(Π G ), α  ∈ Aut(Π G ) a lifting of α, and H → G a connected finite étale Galois subcovering of G  → G such that α  pre- serves the corresponding open subgroup Π H ⊆ Π G , hence induces an element α H ∈ Out(Π H ). Suppose that α H ∈ Dehn(H) [cf. [CbTpI], Definition 4.4]. Then α ∈ Dehn(G). Proof. It follows immediately from [CmbGC], Propositions 1.2, (ii); 1.5, (ii), that α ∈ Aut(G). The fact that α ∈ Dehn(G) now follows from [CmbGC], Propositions 1.2, (i); 1.5, (i), together with the commensu- rable terminality of VCN-subgroups of Π G [cf. [CmbGC], Proposition 1.2, (ii)] and the slimness of verticial subgroups of Π G [cf. [CmbGC], 14 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Remark 1.1.3]. [Here, we recall that an automorphism of a slim profi- nite group is equal to the identity if and only if it preserves and induces the identity on an open subgroup.]  Lemma 1.5 (Outer representations of VA-, NN-, PIPSC-type and finite étale coverings). In the notation of Definition 1.3, sup- † pose that I is isomorphic to Z Σ as an abstract profinite group; let ρ  J : J → Aut(Π G ) be a lifting of the restriction of ρ to an open subgroup J ⊆ I and H → G a connected finite étale Galois subcovering of G  → G such that the action of J on Π G , via ρ  J , preserves the corresponding open subgroup Π H ⊆ Π G , hence induces a continuous homomorphism J → Aut(Π H ). Then ρ is of VA-type [cf. [NodNon], Definition 2.4, (ii), as well as Remark 1.5.1 below] (respectively, NN-type [cf. [NodNon], Definition 2.4, (iii)]; PIPSC-type [cf. Definition 1.3]) if and only if the composite J → Aut(Π H )  Out(Π H ) is of VA-type (respectively, NN-type; PIPSC-type). Proof. Necessity in the case of outer representations of VA-type (re- spectively, NN-type; PIPSC-type) follows immediately from [NodNon], Lemma 2.6, (i) (respectively, [NodNon], Lemma 2.6, (i); the various definitions involved, together with the well-known properness of the moduli stack of pointed stable curves of a given type). To verify suf- ficiency, let us first observe that it follows immediately from the vari- ous definitions involved that we may assume without loss of generality that J = I, and that the outer representation J = I → Out(Π H ) is of SVA-type (respectively, SNN-type; IPSC-type) [cf. [NodNon], Def- inition 2.4]. Then sufficiency in the case of outer representations of VA-type (respectively, NN-type; PIPSC-type) follows immediately, in light of the criterion of [CbTpI], Corollary 5.9, (i) (respectively, (ii); (iii)), from Lemma 1.4, together with the compatibility property of [CbTpI], Corollary 5.9, (v) [applied, via [CbTpI], Theorem 4.8, (ii), (iv), to each of the Dehn coordinates of the profinite Dehn multi-twists under consideration — cf. the proof of [CbTpII], Lemma 3.26, (ii)]. This completes the proof of Lemma 1.5.  Remark 1.5.1. Here, we take the opportunity to correct an unfor- tunate misprint in [NodNon]. The phrase “of VA-type” that appears near the beginning of [NodNon], Definition 2.4, (ii), should read “is of VA-type”. Definition 1.6. Let H be a semi-graph of anabelioids of pro-Σ † PSC- type. Write H for the underlying semi-graph of H, Π H for the [pro-Σ † ]  → H for the universal covering of H fundamental group of H, and H COMBINATORIAL ANABELIAN TOPICS III 15 corresponding to Π H . Let Π ∗G (respectively, Π ∗H ) be a maximal almost pro-Σ quotient of Π G (respectively, Π H ) [cf. Definition 1.1]. (i) For each v ∈ Vert(G) (respectively, e ∈ Edge(G); e ∈ Node(G); e ∈ Cusp(G); z ∈ VCN(G)), we shall refer to the image of a ver- ticial (respectively, an edge-like; a nodal; a cuspidal; a VCN- [cf. [CbTpI], Definition 2.1, (i)]) subgroup of Π G associated to v (respectively, e; e; e; z) in the quotient Π ∗G as a verticial (respectively, an edge-like; a nodal; a cuspidal; a VCN-) sub- group of Π ∗G associated to v (respectively, e; e; e; z). For each  (respectively, e  ∈ Edge( G);  e  ∈ Node( G);  element v  ∈ Vert( G)  z  ∈ VCN( G)),  we shall refer to the image of the e  ∈ Cusp( G); verticial (respectively, edge-like; nodal; cuspidal; VCN-) sub- group of Π G associated to v  (respectively, e  ; e  ; e  ; z  ) in the quotient Π ∗G as the verticial (respectively, edge-like; nodal; cus- pidal; VCN-) subgroup of Π ∗G associated to v  (respectively, e  ; e  ; e  ; z  ). ∼ (ii) We shall say that an isomorphism Π ∗G → Π ∗H is group-theoreti- cally verticial (respectively, group-theoretically nodal; group- theoretically cuspidal) if the isomorphism induces a bijection between the set of the verticial (respectively, nodal; cuspidal) subgroups [cf. (i)] of Π ∗G and the set of the verticial (respec- tively, nodal; cuspidal) subgroups of Π ∗H . We shall say that ∼ an outer isomorphism Π ∗G → Π ∗H is group-theoretically verti- cial (respectively, group-theoretically nodal; group-theoretically cuspidal) if the outer isomorphism arises from an isomorphism ∼ ∗ Π ∗G → Π H which is group-theoretically verticial (respectively, group-theoretically nodal; group-theoretically cuspidal). ∼ (iii) We shall say that an isomorphism Π ∗G → Π ∗H is group-theoreti- cally graphic if the isomorphism is group-theoretically verti- cial, group-theoretically nodal, and group-theoretically cuspi- ∼ dal [cf. (ii)]. We shall say that an outer isomorphism Π ∗G → Π ∗H is group-theoretically graphic if the outer isomorphism arises ∼ from an isomorphism Π ∗G → Π ∗H which is group-theoretically graphic. We shall write Aut grph (Π ∗G ) ⊆ Aut(Π ∗G ) for the subgroup of group-theoretically graphic automorphisms of Π ∗G and Out grph (Π ∗G ) = Aut grph (Π ∗G )/Inn(Π ∗G ) ⊆ Out(Π ∗G ) def for the subgroup of group-theoretically graphic outomorphisms of Π ∗G . 16 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (iv) Let I be a profinite group. Then we shall say that a continuous homomorphism ρ : I → Aut grph (Π ∗G ) ⊆ Aut(Π ∗G ) [cf. (iii)] is of VA-type (respectively, NN-type; PIPSC-type) if the following condition is satisfied: Let N ⊆ Π G be a normal open subgroup of Π G with respect to which Π ∗G is the maximal almost pro- Σ quotient of Π G . [Thus, N Σ ⊆ Π ∗G .] Then there exists a characteristic open subgroup M ⊆ Π ∗G of Π ∗G such that the following conditions are satisfied: (1) M ⊆ N Σ . [Thus, M may be regarded as the [pro-Σ] funda- Σ — cf. [SemiAn], mental group of the pro-Σ completion G M Definition 2.9, (ii) — of the connected finite étale Galois subcovering G M → G of G  → G corresponding to M ⊆ Π ∗G , Σ .] i.e., M = Π G M Σ ), (2) The composite I → Aut(M )  Out(M ) = Out(Π G M where the first arrow is the homomorphism induced by ρ, is of VA-type (respectively, NN-type; PIPSC-type) in the sense of [NodNon], Definition 2.4, (ii) [cf. also Re- mark 1.5.1 of the present paper] (respectively, [NodNon], Definition 2.4, (iii); Definition 1.3 of the present paper) [i.e., as an outer representation of pro-Σ PSC-type — cf. [NodNon], Definition 2.1, (i)]. [Here, we observe that it follows immediately from Lemma 1.5 that condition (2) is independent of the choice of M — cf. Lemma 1.9 below.] We shall say that a continuous homomor- phism ρ : I → Out grph (Π ∗G ) ⊆ Out(Π ∗G ) [cf. (iii)] is of VA-type (respectively, NN-type; PIPSC-type) if ρ arises from a homo- morphism I → Aut grph (Π ∗G ) ⊆ Aut(Π ∗G ) which is of VA-type (respectively, NN-type; PIPSC-type). [Here, we observe that it follows immediately from Lemma 1.5, together with the slim- ness of Π ∗G [cf. Proposition 1.7, (i), below], that this condition on ρ : I → Out grph (Π ∗G ) is independent of the choice of the ho- momorphism I → Aut grph (Π ∗G ).] (v) Let α ∈ Out(Π ∗G ). Then we shall say that α is a profinite Dehn  there exists a lifting multi-twist of Π ∗G if, for each v  ∈ Vert( G), ∗ α[ v ] ∈ Aut(Π G ) of α which preserves the verticial subgroup ∗  [cf. (i)] and induces the Π v  ⊆ Π ∗G associated to v  ∈ Vert( G) ∗ identity automorphism of Π v  . We shall write Dehn(Π ∗G ) ⊆ Out(Π ∗G ) for the subgroup of profinite Dehn multi-twists of Π ∗G . COMBINATORIAL ANABELIAN TOPICS III 17 Remark 1.6.1. In the notation of Definition 1.6, if Π ∗G , Π ∗H are the respective maximal almost pro-Σ quotients of Π G , Π H with respect to Π G , Π H , then it follows immediately from the various definitions involved that Π ∗G , Π ∗H are the respective maximal pro-Σ quotients of Π G , Π H . In particular, it follows immediately that one may regard Π ∗G , Π ∗H as the [pro-Σ] fundamental groups of the semi-graphs of anabelioids of pro-Σ PSC-type G Σ , H Σ obtained by forming the pro-Σ completions [cf. [SemiAn] Definition 2.9, (ii)] of G, H, respectively, i.e., Π ∗G = Π G Σ , Π ∗H = Π H Σ . Moreover, one verifies immediately that, relative to these identifications, the notions defined in Definition 1.6, (i), (ii), (iii), (iv), are compatible with their counterparts defined [for the most part] in earlier papers of the authors: • VCN-subgroups [cf. [CbTpI], Definition 2.1, (i)]; • group-theoretically verticial/nodal/cuspidal/graphic (outer) iso- morphisms [cf. [CmbGC], Definition 1.4, (i), (iv); [NodNon], Definition 1.12]; • outer representations of VA-/NN-/PIPSC-type [cf. [NodNon], Definition 2.4, (ii), (iii); Remark 1.5.1 of the present paper; Definition 1.3 of the present paper; Lemma 1.5 of the present paper]; • profinite Dehn multi-twists [cf. [CbTpI], Definition 4.4], i.e., so Dehn(G Σ ) = Dehn(Π ∗G ) ⊆ Out grph (Π ∗G ). Remark 1.6.2. In the situation of Definition 1.6, (iv), it follows imme- diately from Lemma 1.5, together with [NodNon], Remark 2.4.2, that we have implications PIPSC-type =⇒ NN-type =⇒ VA-type . Proposition 1.7 (Properties of VCN-subgroups). Let Π ∗G be a maximal almost pro-Σ quotient of Π G [cf. Definition 1.1]. For v  ,  e  ∈ Edge( G),  write G ∗ → G for the connected profinite w  ∈ Vert( G); étale subcovering of G  → G corresponding to Π ∗G ; Vert(G ∗ ) = lim Vert(G  ), Edge(G ∗ ) = lim Edge(G  ) ←− ←− — where the projective limits range over all connected finite étale sub- coverings G  → G of G ∗ → G; def def v  (G ∗ ) ∈ Vert(G ∗ ), e  (G ∗ ) ∈ Edge(G ∗ )  e  ∈ Edge( G)  via the natural maps for the images of v  ∈ Vert( G), ∗ ∗   Vert( G)  Vert(G ), Edge( G)  Edge(G ), respectively; ∗ E G ∗ : Vert(G ∗ ) −→ 2 Edge(G ) 18 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [cf. the discussion entitled “Sets” in [CbTpI], §0, concerning the no- ∗ tation 2 Edge(G ) ] for the map induced by the various E’s involved [cf. [NodNon], Definition 1.1, (iv)]; δ( v (G ∗ ), w(G  ∗ )) = sup {δ( v (G  ), w(G   ))} ∈ N ∪ {∞} def G  [cf. [NodNon], Definition 1.1, (vii)] — where G  ranges over the con- nected finite étale subcoverings G  → G of G ∗ → G. Then the following hold: (i) Π ∗G is topologically finitely generated, slim [cf. the discus- sion entitled “Topological groups” in [CbTpI], §0], and almost torsion-free [cf. the discussion entitled “Topological groups” in [CbTpI], §0]. In particular, every VCN-subgroup of Π ∗G [cf. Definition 1.6, (i)] is almost torsion-free. (ii) Let z ∈ VCN(G) and Π z ⊆ Π G a VCN-subgroup of Π G associ- ated to z ∈ VCN(G). Write Π ∗ z ⊆ Π G ∗ for the VCN-subgroup of Π ∗G obtained by forming the image of Π z ⊆ Π G in Π ∗G . Then Π ∗ z is a maximal almost pro-Σ quotient of Π z . In partic- ular, every verticial subgroup of Π ∗G is topologically finitely generated and slim.  Write Π ∗ ⊆ Π ∗ for the verticial (iii) For i = 1, 2, let v  i ∈ Vert( G). G v  i ∗ subgroup of Π G associated to v  i . Consider the following three [mutually exclusive] conditions: (1) δ( v 1 (G ∗ ), v  2 (G ∗ )) = 0. (2) δ( v 1 (G ∗ ), v  2 (G ∗ )) = 1. (3) δ( v 1 (G ∗ ), v  2 (G ∗ )) ≥ 2. Then we have equivalences (1) ⇐⇒ (1  ) ; (2) ⇐⇒ (2  ) ; (3) ⇐⇒ (3  ) with the following three conditions: (1  ) Π ∗ v  1 = Π ∗ v  2 . (2  ) Π ∗ v  1 ∩ Π ∗ v  2 is infinite, but Π ∗ v  1  = Π ∗ v  2 . (3  ) Π ∗ v  1 ∩ Π ∗ v  2 is finite. (iv) In the situation of (iii), if condition (2), hence also condition (2  ), holds, then it holds that (E G ∗ ( v 1 (G ∗ )) ∩ E G ∗ ( v 2 (G ∗ )))  = 1, ∗ ∗ ∗  and, moreover, Π v  1 ∩ Π v  2 = Π e  , for any element e  ∈ Edge( G) ∗ ∗ ∗ such that e  (G ) ∈ E G ∗ ( v 1 (G )) ∩ E G ∗ ( v 2 (G )).  Write Π ∗ ⊆ Π ∗ for the edge-like (v) For i = 1, 2, let e  i ∈ Edge( G). G e  i subgroup of Π ∗G associated to e  i . Then Π ∗ e  1 ∩ Π ∗ e  2 is infinite if COMBINATORIAL ANABELIAN TOPICS III 19 and only if e  1 (G ∗ ) = e  2 (G ∗ ). In particular, Π ∗ e  1 ∩Π ∗ e  2 is infinite if and only if Π ∗ e  1 = Π ∗ e  2 .  e  ∈ Edge( G).  Write Π ∗ , Π ∗ ⊆ Π ∗ for the (vi) Let v  ∈ Vert( G), G v  e  VCN-subgroups of Π ∗G associated to v  , e  , respectively. Then Π ∗ e  ∩ Π ∗ v  is infinite if and only if e  (G ∗ ) ∈ E G ∗ ( v (G ∗ )). In par- ticular, Π ∗ e  ∩ Π ∗ v  is infinite if and only if Π ∗ e  ⊆ Π ∗ v  . (vii) Every VCN-subgroup of Π ∗G is commensurably terminal [cf. the discussion entitled “Topological groups” in [CbTpI], §0] in Π ∗G . (viii) Let z ∈ VCN(G), Π z ⊆ Π G a VCN-subgroup of Π G associated to z ∈ VCN(G), and Π z  Π ‡ z an almost pro-Σ quotient of Π ‡ z . Then there exists a maximal almost pro-Σ quotient Π ∗∗ G of Π G such that the quotient of Π z determined by the quo- ‡ tient Π G  Π ∗∗ G dominates the quotient Π z  Π z [cf. the discussion entitled “Topological groups” in §0]. Proof. Let N ⊆ Π G be a normal open subgroup of Π G with respect to which Π ∗G is the maximal almost pro-Σ quotient of Π G . [Thus, N Σ ⊆ Π ∗G .] Write G N → G for the connected finite étale Galois sub- covering of G  → G corresponding to N ⊆ Π G . Thus, N Σ ⊆ Π ∗G may be regarded as the [pro-Σ] fundamental group of the pro-Σ completion Σ G N [cf. [SemiAn], Definition 2.9, (ii)] of G N , i.e., N Σ = Π G N Σ . First, we verify assertion (i). Since Π G is topologically finitely gen- erated, it is immediate that Π ∗G is topologically finitely generated. Now let us recall [cf. [MzTa], Remark 1.2.2; [MzTa], Proposition 1.4] that N Σ = Π G N Σ is torsion-free and slim. Thus, the fact that Π ∗G is almost torsion-free is immediate; the slimness of Π ∗G follows immediately, by considering the natural outer action Π G /N → Out(N Σ ), from the well- known fact that any nontrivial automorphism of a stable log curve over an algebraically closed field of characteristic ∈ Σ induces a nontrivial outomorphism of the maximal pro-Σ quotient of the geometric log fun- damental group of the stable curve [cf. [CmbGC], Proposition 1.2, (i), (ii); [MzTa], Proposition 1.4, applied to the verticial subgroups of the geometric log fundamental group under consideration]. This completes the proof of assertion (i). Next, we verify assertion (ii). Let us recall that since Π z ∩ N ⊆ N = Π G N is a VCN-subgroup of Π G N , the natural homomorphism (Π z ∩ N ) Σ → Π Σ G N is injective [cf., e.g., the proof of [SemiAn], Proposition 2.5, (i); [SemiAn], Example 2.10]. Thus, it follows immediately from Lemma 1.2, (ii), that Π ∗ z is a maximal almost pro-Σ quotient of Π z . In particular, if z ∈ Vert(G), then it follows immediately from assertion (i) that Π ∗ z is topologically finitely generated and slim. This completes the proof of assertion (ii). 20 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Next, we verify assertions (iii), (v), and (vi). Since N Σ = Π G N Σ , one verifies easily — by considering the intersections of N Σ = Π G N Σ with the various VCN-subgroups of Π ∗G under consideration and ap- plying [NodNon], Lemma 1.9, (ii) (respectively, [NodNon], Lemma 1.5; [NodNon], Lemma 1.7), together with the well-known fact that every VCN-subgroup of Π G N Σ is nontrivial and torsion-free [hence also infinite] — that assertion (iii) (respectively, (v); (vi)) holds. This completes the proof of assertions (iii), (v), and (vi). Assertion (vii) follows formally from assertions (iii), (v). Indeed, let Π ∗ z  ⊆ Π ∗G be the VCN-subgroup  and γ ∈ C Π ∗ (Π ∗ ). Then of Π ∗G associated to an element z  ∈ VCN( G) z  G it follows immediately from assertions (iii), (v) that z  = z  γ ; we thus conclude that γ ∈ Π ∗ z  . This completes the proof of assertion (vii). Next, we verify assertion (iv). By applying [NodNon], Lemma 1.8, Σ to G N , one verifies immediately that (E G ∗ ( v 1 (G ∗ )) ∩ E G ∗ ( v 2 (G ∗ )))  = 1. Thus, it follows immediately from assertion (vi); [NodNon], Lemma 1.5; [NodNon], Lemma 1.9, (ii), that Π ∗ e  ∩N Σ = Π ∗ v  1 ∩Π ∗ v  2 ∩N Σ . Since N Σ is open in Π ∗G , we conclude from assertion (vi) that Π ∗ e  is an open subgroup of Π ∗ v  1 ∩ Π ∗ v  2 , hence that Π ∗ v  1 ∩ Π ∗ v  2 ⊆ C Π ∗G (Π ∗ e  ) = Π ∗ e  [cf. assertion (vii)]. This completes the proof of assertion (iv). Finally, we verify assertion (viii). It follows from the definition of an almost pro-Σ quotient that the natural surjection Π z  Π ‡ z factors through a maximal almost pro-Σ quotient of Π z . Thus, by replacing Π ‡ z by a suitable maximal almost pro-Σ quotient of Π z , we may assume without loss of generality that Π ‡ z is a maximal almost pro-Σ quotient of Π z . Let N z ⊆ Π z be a normal open subgroup of Π z with respect to which Π ‡ z is the maximal almost pro-Σ quotient of Π z and N G ⊆ Π G a normal open subgroup of Π G such that N G ∩ Π z ⊆ N z . Here, we recall that the existence of such a subgroup N G follows immediately from the fact that the natural profinite topology on Π z coincides with the topology on Π z induced by the topology of Π G . Then one verifies easily that the maximal almost pro-Σ quotient of Π G with respect to N G ⊆ Π G is a maximal almost pro-Σ quotient of Π G as in the statement of assertion (viii). This completes the proof of assertion (viii).  Definition 1.8. Let Π ∗G be a maximal almost pro-Σ quotient of Π G [cf. Definition 1.1]. Then we shall write Aut |grph| (Π ∗G ) ⊆ Aut grph (Π ∗G ) for the subgroup of group-theoretically graphic [cf. Definition 1.6, (iii)] automorphisms α of Π ∗G such that the natural action of α on the un- derlying semi-graph G [determined by the group-theoretic graphicity COMBINATORIAL ANABELIAN TOPICS III 21 of α, together with Proposition 1.7, (iii), (v), (vi)] is the identity auto- morphism. Also, we shall write Out |grph| (Π ∗G ) = Aut |grph| (Π ∗G )/Inn(Π ∗G ) ⊆ Out(Π ∗G ) . def for the image of Aut |grph| (Π ∗G ) in Out(Π ∗G ). Remark 1.8.1. In the notation of Definition 1.8, one verifies easily that Dehn(Π ∗G ) ⊆ Out |grph| (Π ∗G ) [cf. Definitions 1.6, (v); 1.8; [CmbGC], Proposition 1.2, (i)]. Remark 1.8.2. In the spirit of Remark 1.6.1, one verifies immediately that the notation of Definition 1.8 is consistent with the the notation of [CbTpI], Definition 2.6, (i) [cf. also [CbTpII], Remark 4.1.2]. Lemma 1.9 (Alternative characterization of outer representa- tions of VA-, NN-, PIPSC-type). Let Π ∗G be a maximal almost pro-Σ quotient of Π G [cf. Definition 1.1], I a profinite group, and ρ : I → Aut grph (Π ∗G ) a continuous homomorphism. Then the following conditions are equivalent: (i) ρ is of VA-type (respectively, NN-type; PIPSC-type) [cf. Definition 1.6, (iv)]. (ii) Let N ⊆ Π G be a normal open subgroup of Π G with respect to which Π ∗G is the maximal almost pro-Σ quotient of Π G . [Thus, N Σ ⊆ Π ∗G .] Let M ⊆ Π ∗G be a characteristic open subgroup of Π ∗G such that M ⊆ N Σ . [Thus, M may be regarded as Σ the [pro-Σ] fundamental group of the pro-Σ completion G M — cf. [SemiAn], Definition 2.9, (ii) — of the connected finite étale Galois subcovering G M → G of G  → G corresponding to Σ .] M ⊆ Π ∗G , i.e., M = Π G M Then it holds that the compos- Σ ) ite of the resulting homomorphism I → Aut(M ) = Aut(Π G M Σ ) → Out(Π G Σ ) is an outer with the natural projection Aut(Π G M M representation of VA-type (respectively, NN-type; PIPSC- type) in the sense of [NodNon], Definition 2.4, (ii) [cf. also Remark 1.5.1 of the present paper] (respectively, [NodNon], Definition 2.4, (iii); Definition 1.3 of the present paper). Proof. The implication (ii) ⇒ (i) is immediate; the implication (i) ⇒ (ii) follows immediately from Lemma 1.5. This completes the proof of Lemma 1.9.  22 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Lemma 1.10 (Automorphisms of semi-graphs of anabelioids of PSC-type with prescribed underlying semi-graphs). Let Π ∗G be a maximal almost pro-Σ quotient of Π G [cf. Definition 1.1] and α ∈ Out(Π ∗G ). Suppose that there exist distinct elements v 1 , v 2 , v 3 ∈ Vert(G); e 1 , e 2 ∈ Node(G) such that Vert(G) = {v 1 , v 2 , v 3 }; Node(G) = {e 1 , e 2 }; V(e i ) = {v i , v i+1 } [where i ∈ {1, 2}]. For each i ∈ {1, 2}, write Π G ∗ {e } for the maximal almost pro-Σ quotient of i Π G {ei } [cf. [CbTpI], Definition 2.8] determined by the natural outer ∼ isomorphism Φ G {ei } : Π G {ei } → Π G [cf. [CbTpI], Definition 2.10] and ∼ the maximal almost pro-Σ quotient Π ∗G of Π G ; Φ ∗G {e } : Π ∗G {e } → Π ∗G i i for the outer isomorphism determined by Φ G {ei } . Suppose, moreover, that, for each i ∈ {1, 2}, the outomorphism of Π ∗G {e } obtained by con- i jugating α by Φ ∗G {e } is a profinite Dehn multi-twist of Π ∗G {e } [cf. i i Definition 1.6, (v)]. Then α is the identity outomorphism. Proof. First, let us observe that it follows immediately from the defi- nition of a profinite Dehn multi-twist that α is a profinite Dehn multi- twist of Π ∗G . Let us fix a verticial subgroup Π v 2 ⊆ Π G associated to v 2 . Let Π e 1 , Π e 2 ⊆ Π G be nodal subgroups associated to e 1 , e 2 , respectively, which are contained in Π v 2 ; Π v 1 ⊆ Π G a verticial subgroup associated to v 1 which contains Π e 1 ; Π v 3 ⊆ Π G a verticial subgroup associated to v 3 which contains Π e 2 . Thus, we have inclusions Π v 1 ⊇ Π e 1 ⊆ Π v 2 ⊇ Π e 2 ⊆ Π v 3 . For each z ∈ {v 1 , v 2 , v 3 , e 1 , e 2 }, write Π ∗ z ⊆ Π ∗G for the VCN-subgroup of Π ∗G associated to z obtained by forming the image of Π z ⊆ Π G in Π ∗G . Then since α is a profinite Dehn multi-twist, there exists a lifting α[v 2 ] ∈ Aut(Π ∗G ) of α which preserves and induces the identity auto- morphism of Π ∗ v 2 ; in particular, α[v 2 ] preserves and induces the identity automorphisms of Π ∗ e 1 , Π ∗ e 2 . Moreover, by applying a similar argument to the argument given in the proof of [CbTpI], Lemma 4.6, (i), where we replace [CmbGC], Remark 1.1.3 (respectively, [CmbGC], Proposi- tion 1.2, (ii); [CbTpI], Proposition 4.5; [NodNon], Lemma 1.7), in the proof of [CbTpI], Lemma 4.6, (i), by Proposition 1.7, (ii) (respectively, Proposition 1.7, (vii); Remark 1.8.1; Proposition 1.7, (vi)), we conclude that α[v 2 ](Π ∗ v 1 ) = Π ∗ v 1 , α[v 2 ](Π ∗ v 3 ) = Π ∗ v 3 , and, moreover, that there ex- ist unique elements γ 1 ∈ Π ∗ e 1 , γ 2 ∈ Π ∗ e 2 such that the restrictions of α[v 2 ] to Π ∗ v 1 , Π ∗ v 3 are the inner automorphisms determined by γ 1 , γ 2 , respectively. Thus, since, for each i ∈ {1, 2}, the outomorphism of Π ∗G {e } obtained by conjugating α by Φ ∗G {e } is a profinite Dehn multi- i i twist of Π ∗G {e } , one verifies easily — by considering the restriction of i this outomorphism of Π ∗G {e } to the unique conjugacy class of verticial i subgroups of Π ∗G {e } that does not arise from a conjugacy class of ver- i ticial subgroups of Π ∗G [cf. also Proposition 1.7, (ii), (vii)] — that γ 1 COMBINATORIAL ANABELIAN TOPICS III 23 and γ 2 are trivial. On the other hand, it follows immediately from a similar argument to the argument applied in the proof of [CmbCsp], Proposition 1.5, (iii), that Π G is topologically generated by Π v 1 , Π v 2 , and Π v 3 ; in particular, Π ∗G is topologically generated by Π ∗ v 1 , Π ∗ v 2 , and Π ∗ v 3 . Thus, we conclude that α[v 2 ] is the identity automorphism of Π ∗G . This completes the proof of Lemma 1.10.  Theorem 1.11 (Group-theoretic verticiality/nodality of iso- morphisms of outer representations of NN-, PIPSC-type). Let Σ ⊆ Σ † be nonempty sets of prime numbers, G (respectively, H) a semi-graph of anabelioids of pro-Σ † PSC-type, Π G (respectively, Π H ) the [pro-Σ † ] fundamental group of G (respectively, H), Π ∗G (respectively, Π ∗H ) a maximal almost pro-Σ quotient [cf. Definition 1.1] of Π G ∼ (respectively, Π H ), α : Π ∗G → Π ∗H an isomorphism of profinite groups, I (respectively, J) a profinite group, ρ I : I → Out grph (Π ∗G ) (respectively, ρ J : J → Out grph (Π ∗H )) [cf. Definition 1.6, (iii)] a continuous homo- ∼ morphism, and β : I → J an isomorphism of profinite groups. Suppose that the diagram ρ I → Out(Π ∗G ) I −−− ⏐ ⏐ ⏐ ⏐ Out(α) β   ρ J J −−− → Out(Π ∗H ) — where the right-hand vertical arrow is the isomorphism obtained by conjugating by α — commutes. Then the following hold: (i) Suppose, moreover, that ρ I , ρ J are of NN-type [cf. Defini- tion 1.6, (iv)]. Then the following three conditions are equiva- lent: (1) The isomorphism α is group-theoretically verticial [i.e., roughly speaking, preserves verticial subgroups — cf. Def- inition 1.6, (ii)]. (2) The isomorphism α is group-theoretically nodal [i.e., roughly speaking, preserves nodal subgroups — cf. Defini- tion 1.6, (ii)]. (3) There exists an infinite subgroup H ⊆ Π ∗G of Π ∗G such that H ⊆ Π ∗G , α(H) ⊆ Π ∗H are contained in verticial subgroups of Π ∗G , Π ∗H , respectively [cf. Definition 1.6, (i)]. (ii) Suppose, moreover, that ρ I is of NN-type, and that ρ J is of PIPSC-type [cf. Definition 1.6, (iv)]. [For example, this will be the case if both ρ I and ρ J are of PIPSC-type — cf. 24 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Remark 1.6.2.] Then α is group-theoretically verticial, hence also group-theoretically nodal. Proof. The implication (1) ⇒ (2) of assertion (i) and the final portion of assertion (ii) [i.e., the portion concerning group-theoretic nodality] fol- low immediately from Proposition 1.7, (iv). The implication (2) ⇒ (3) of assertion (i) is immediate. Finally, we verify assertion (ii) (respec- tively, the implication (3) ⇒ (1) of assertion (i)). Suppose that ρ I , ρ J are as in assertion (ii) (respectively, condition (3) of assertion (i)). Let N G ⊆ Π G , N H ⊆ Π H be normal open subgroups of Π G , Π H with respect to which Π ∗G , Π ∗H are the maximal almost pro-Σ quotients of Π G , Π H , respectively. [Thus, N G Σ ⊆ Π ∗G , N H Σ ⊆ Π ∗H .] Now it follows immediately from the fact that Π ∗G , Π ∗H are topologically finitely gen- erated [cf. Proposition 1.7, (i)] that there exists a characteristic open def subgroup M G ⊆ Π ∗G of Π ∗G such that M G ⊆ N G Σ , M H = α(M G ) ⊆ N H Σ . Thus, it follows immediately, in light of Lemma 1.9, from [CbTpII], Theorem 1.9, (ii) (respectively, the implication (3) ⇒ (1) of [CbTpII], Theorem 1.9, (i)), together with Proposition 1.7, (vii), that α is group- theoretically verticial. This completes the proof of Theorem 1.11.  Corollary 1.12 (Group-theoretic graphicity of group-theoreti- cally cuspidal isomorphisms of outer representations of NN-, PIPSC-type). Let Σ ⊆ Σ † be nonempty sets of prime numbers, G (respectively, H) a semi-graph of anabelioids of pro-Σ † PSC-type, Π G (respectively, Π H ) the [pro-Σ † ] fundamental group of G (respectively, H), Π ∗G (respectively, Π ∗H ) a maximal almost pro-Σ quotient [cf. ∼ Definition 1.1] of Π G (respectively, Π H ), α : Π ∗G → Π ∗H an isomor- phism of profinite groups, I (respectively, J) a profinite group, ρ I : I → Out grph (Π ∗G ) (respectively, ρ J : J → Out grph (Π ∗H )) [cf. Definition 1.6, ∼ (iii)] a continuous homomorphism, and β : I → J an isomorphism of profinite groups. Suppose that the following conditions are satisfied: (i) The diagram ρ I I −−− → Out(Π ∗G ) ⏐ ⏐ ⏐ ⏐ Out(α) β   ρ J J −−− → Out(Π ∗H ) — where the right-hand vertical arrow is the isomorphism ob- tained by conjugating by α — commutes. (ii) α is group-theoretically cuspidal [cf. Definition 1.6, (ii)]. (iii) ρ I , ρ J are of NN-type [cf. Definition 1.6, (iv)]. Suppose, moreover, that one of the following conditions is satisfied: COMBINATORIAL ANABELIAN TOPICS III 25 (1) Cusp(G)  = ∅. (2) Either ρ I or ρ J is of PIPSC-type [cf. Definition 1.6, (iv)]. Then α is group-theoretically graphic [cf. Definition 1.6, (iii)]. Proof. This follows immediately from Theorem 1.11, (i) (respectively, (ii)), whenever condition (1) (respectively, (2)) is satisfied.  26 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI 2. Almost pro-Σ injectivity In the present §2, we develop an almost pro-Σ version of the injec- tivity portion of the theory of combinatorial cuspidalization [cf. The- orem 2.9, Corollary 2.10 below]. We also discuss an almost pro-l analogue [cf. Corollary 2.13 below] of the tripod homomorphism of [CbTpII], Definition 3.19. The significance of developing these almost pro-l analogues of standard results in combinatorial anabelian geome- try is that they allow one to apply techniques that are, a priori, only available in the pro-l case to profinite fundamental groups. Such pro-l techniques will be necessary to verify the results obtained in §3 concern- ing various metric aspects of the outer representations of Galois groups that arise from hyperbolic curves and their associated configuration spaces over p-adic fields. In the present §2, let Σ be a nonempty set of prime numbers. Definition 2.1. Let l be a prime number; n a positive integer; (g, r) a pair of nonnegative integers such that 2g − 2 + r > 0; k an algebraically closed field of characteristic zero; (Spec k) log the log scheme obtained by equipping Spec k with the log structure determined by the fs chart N → k that maps 1 → 0; X log a stable log curve of type (g, r) over (Spec k) log . For each positive integer i, write X i log for the i-th log configuration space of X log [cf. the discussion entitled “Curves” in [CbTpII], §0]; Π i for the pro-Primes configuration space group [cf. [ExtFam], Theorem B; [MzTa], Definition 2.3, (i)] given by the kernel of the natural outer surjection π 1 (X i log )  π 1 ((Spec k) log ). Let Π n  Π n ∗ be a quotient of Π n . Write {1} = Π n/n ⊆ Π n/n−1 ⊆ · · · ⊆ Π n/m ⊆ · · · ⊆ Π n/2 ⊆ Π n/1 ⊆ Π n/0 = Π n for the standard fiber filtration on Π n — i.e., Π n/m ⊆ Π n is the kernel of some fixed surjection [that belongs to the collection of surjections that constitutes the outer surjection] p Π n/m : Π n  Π m induced by the log log obtained by forgetting the factors labeled projection p n/m : X n log → X m by indices > m [cf. [CmbCsp], Definition 1.1, (i)]; ∗ ⊆ · · · ⊆ Π ∗ n/2 ⊆ Π ∗ n/1 ⊆ Π ∗ n/0 = Π ∗ n {1} = Π ∗ n/n ⊆ Π ∗ n/n−1 ⊆ · · · ⊆ Π n/m for the induced filtration on Π ∗ n . (i) For each 1 ≤ m ≤ n, we shall refer to the subquotient Π ∗ n/m−1 /Π ∗ n/m of Π ∗ n as a standard-adjacent subquotient of Π ∗ n . (ii) We shall say that Π ∗ n is an SA-maximal almost pro-l quo- tient of Π n [where the “SA” stands for “standard-adjacent”] if, for every 1 ≤ m ≤ n, the natural quotient Π n/m−1 /Π n/m  Π ∗ n/m−1 /Π ∗ n/m is a maximal almost pro-l quotient of Π n/m−1 /Π n/m [cf. Definition 1.1]. COMBINATORIAL ANABELIAN TOPICS III 27 (iii) We shall say that Π ∗ n is F-characteristic if every F-admissible automorphism [cf. [CmbCsp], Definition 1.1, (ii)] of Π n pre- serves the kernel of the quotient Π n  Π ∗ n . (iv) We shall refer to the image of a fiber subgroup [cf. [MzTa], Definition 2.3, (iii)] of Π n in Π ∗ n as a fiber subgroup of Π ∗ n . For each 1 ≤ m ≤ n, we shall refer to the image of a cuspidal inertia subgroup of Π n/m−1 /Π n/m in Π ∗ n/m−1 /Π ∗ n/m as a cuspidal inertia subgroup of Π ∗ n/m−1 /Π ∗ n/m . (v) Let α be an automorphism of Π ∗ n . Then we shall say that α is F-admissible if α preserves each fiber subgroup [cf. (iv)] of Π ∗ n . We shall say that α is C-admissible if α preserves the filtration {1} = Π ∗ n/n ⊆ Π ∗ n/n−1 ⊆ · · · ⊆ Π ∗ n/m ⊆ · · · ⊆ Π ∗ n/2 ⊆ Π ∗ n/1 ⊆ Π ∗ n/0 = Π ∗ n , and, moreover, α induces a bijection of the set of cuspidal iner- tia subgroups [cf. (iv)] of every standard-adjacent subquotient [cf. (i)] of Π ∗ n . We shall say that α is FC-admissible if α is F-admissible and C-admissible. (vi) Let α be an outomorphism of Π ∗ n . Then we shall say that α is F-admissible (respectively, C-admissible; FC-admissible) if α arises from an automorphism of Π ∗ n that is F-admissible (respectively, C-admissible; FC-admissible) [cf. (v)]. (vii) Write Aut F (Π ∗ n ) , Aut C (Π ∗ n ) , Aut FC (Π ∗ n ) ⊆ Aut(Π ∗ n ) for the respective subgroups of F-, C-, and FC-admissible au- tomorphisms of Π ∗ n [cf. (v)]; Out F (Π ∗ n ) = Aut F (Π ∗ n )/Inn(Π ∗ n ) , def Out C (Π ∗ n ) = Aut C (Π ∗ n )/Inn(Π ∗ n ) , def Out FC (Π ∗ n ) = Aut FC (Π ∗ n )/Inn(Π ∗ n ) ⊆ Out(Π ∗ n ) for the respective subgroups of F-, C-, and FC-admissible out- omorphisms of Π ∗ n [cf. (vi)]. def (viii) Let Π n  Π ∗∗ n be a quotient of Π n that dominates the quotient Π n  Π ∗ n [cf. the discussion entitled “Topological groups” in §0]. Then we shall write F ∗ ∗∗ Out F (Π ∗∗ n  Π n ) ⊆ Out (Π n ) , FC ∗ ∗∗ Out FC (Π ∗∗ n  Π n ) ⊆ Out (Π n ) [cf. (vii)] for the respective subgroups of F-, FC-admissible outomorphisms of Π ∗∗ n that preserve the kernel of the natural ∗  Π . surjection Π ∗∗ n n Thus, we have natural homomorphisms F ∗ ∗ Out F (Π ∗∗ n  Π n ) −→ Out (Π n ) , 28 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI FC ∗ ∗ Out FC (Π ∗∗ n  Π n ) −→ Out (Π n ) . We shall write F ∗ Out F (Π ∗ n  Π ∗∗ n ) ⊆ Out (Π n ) , FC ∗ Out FC (Π ∗ n  Π ∗∗ n ) ⊆ Out (Π n ) for the respective images of these natural homomorphisms. Thus, we have natural surjections F ∗ ∗ ∗∗ Out F (Π ∗∗ n  Π n )  Out (Π n  Π n ) , FC ∗ ∗ ∗∗ Out FC (Π ∗∗ n  Π n )  Out (Π n  Π n ) . Remark 2.1.1. In the notation of Definition 2.1, suppose that Π ∗ n is F-characteristic [cf. Definition 2.1, (iii)]. Then it follows from the various definitions involved that Out F (Π n  Π ∗ n ) = Out F (Π n ), Out FC (Π n  Π ∗ n ) = Out FC (Π n ) [cf. Definition 2.1, (viii)]; thus, we have natural surjections Out F (Π n )  Out F (Π ∗ n  Π n ), Out FC (Π n )  Out FC (Π ∗ n  Π n ) [cf. Definition 2.1, (viii)]. Lemma 2.2 (Preservation of quotients of extensions). Let 1 −−−→ N −−−→ G −−−→ Q −−−→ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐    1 −−−→ N −−−→ G −−−→ Q −−−→ 1 be a commutative diagram of profinite groups — where the horizontal sequences are exact, and the vertical arrows are surjective. Write G ∗ = Ker(G  Q  Q)/Ker(N  N ) def and N ∗ for the image of N in G ∗ . Suppose that N is center-free. Then the image of Ker(G  G) in G ∗ is equal to the centralizer Z G ∗ (N ∗ ). Proof. Observe that, by replacing G by Ker(G  Q  Q) (= G × Q Ker(Q  Q)), we may assume without loss of generality that Q = {1}. In a similar vein, by replacing G by G/Ker(N  N ), we may assume without loss of generality that N = N , which [since Q = {1}] implies that G = G ∗ , N = N ∗ = N . Then one verifies easily that the natural inclusions N , Ker(G  G) → G determine an isomorphism ∼ N × Ker(G  G) → G. Thus, since N is center-free, we obtain that  Ker(G  G) = Z G (N ). This completes the proof of Lemma 2.2. COMBINATORIAL ANABELIAN TOPICS III 29 Proposition 2.3 (Existence of F-characteristic SA-maximal al- most pro-l quotients). In the notation of Definition 2.1, let Π n  Π ∗ n be a quotient of Π n . Then the following hold: (i) If Π ∗ n is an SA-maximal almost pro-l quotient of Π n [cf. Definition 2.1, (ii)], then Π ∗ n is topologically finitely gen- erated, almost pro-l [cf. the discussion entitled “Topological groups” in [CbTpI], §0], and slim [cf. the discussion entitled “Topological groups” in [CbTpI], §0]. (ii) Let 0 ≤ m 1 ≤ m 2 ≤ n be integers and (Π n/m 1 /Π n/m 2 ) ‡ an almost pro-l quotient of Π n/m 1 /Π n/m 2 . Then there exists an F-characteristic [cf. Definition 2.1, (iii)] SA-maximal almost pro-l quotient Π ∗∗ n of Π n such that the quotient of Π n/m 1 /Π n/m 2 determined by the quotient Π n  Π ∗∗ n domi- nates the quotient Π n/m 1 /Π n/m 2  (Π n/m 1 /Π n/m 2 ) ‡ [cf. the discussion entitled “Topological groups” in §0]. (iii) Let 1 ≤ m ≤ n be an integer, H ⊆ Π n/m−1 /Π n/m a VCN- subgroup of Π n/m−1 /Π n/m [cf. [CbTpII], Definition 3.1, (iv)], and H  H ‡ an almost pro-l quotient of H. Then there exists an F-characteristic SA-maximal almost pro-l quo- tient Π ∗∗ n of Π n such that the quotient of H determined by the ‡ quotient Π n  Π ∗∗ n dominates the quotient H  H . Proof. First, we verify assertion (i). Observe that it follows imme- diately from Proposition 1.7, (i), together with the definition of an SA-maximal almost pro-l quotient, that Π ∗ n is a successive extension of almost pro-l, topologically finitely generated, slim profinite groups. Thus, one verifies immediately that Π ∗ n is almost pro-l, topologically finitely generated, and slim. This completes the proof of assertion (i). Next, we verify assertion (ii). First, observe that since (Π n/m 1 /Π n/m 2 ) ‡ may be regarded as an almost pro-l quotient of Π n/m 1 , we may assume def without loss of generality that m 2 = n. Write m = m 1 . If m = n, then one may take the quotient Π ∗∗ n to be the maximal pro-l quotient of Π n [cf. [MzTa], Proposition 2.2, (i)]. Thus, we may assume without loss of generality that m ≤ n − 1. Let us verify assertion (ii) by induction on n. If n = 1, then as- sertion (ii) follows immediately from the fact that Π 1 is topologically finitely generated, which implies that the topology of Π 1 admits a basis of characteristic open subgroups. Thus, we suppose that n ≥ 2, and that the induction hypothesis is in force. Then observe that since the subgroup Π n/n−1 ⊆ Π n may be regarded as the “Π 1 ” associated to some stable log curve of type (g, r + n − 1), by applying the induction hy- pothesis to the quotient Π n/n−1  Π ‡ n/n−1 determined by the quotient Π n/m  Π ‡ n/m , we obtain an F-characteristic SA-maximal almost pro-l 30 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ‡ quotient Π ∗∗ n/n−1 of Π n/n−1 which dominates Π n/n−1  Π n/n−1 . In par- ticular, since the quotient Π n/n−1  Π ∗∗ n/n−1 is F-characteristic, hence arises from a subgroup of Π n/n−1 which is normal in Π n , we thus obtain a natural outer action ∼ Π n /Π n/n−1 ( → Π n−1 ) → Out(Π ∗∗ n/n−1 ) . Since the profinite group Π ∗∗ n/n−1 is almost pro-l and topologically finitely generated [cf. assertion (i)], it follows immediately that the outer action Π n /Π n/n−1 → Out(Π ∗∗ n/n−1 ) factors through an almost pro-l quotient Π n /Π n/n−1  Q of Π n /Π n/n−1 . In particular, it follows that the natural outer action Π n/m /Π n/n−1 ⊆ Π n /Π n/n−1 → Out(Π ∗∗ n/n−1 ) factors through an al- most pro-l quotient of Π n/m /Π n/n−1 . Note that this implies [cf. the slimness of Π ∗∗ n/n−1 proved in assertion (i)] that there exists an al- most pro-l quotient Π n/m  Q ∗∗ of Π n/m that induces the quotient Π n/n−1  Π ∗∗ n/n−1 of Π n/n−1 . Now one verifies immediately that the ∗∗∗ quotient Q determined by the intersection of the kernels of the two quotients Π n/m  Π ‡ n/m , Π n/m  Q ∗∗ is an almost pro-l quotient of Π n/m that induces the quotient Π n/n−1  Π ∗∗ n/n−1 of Π n/n−1 . Thus, we conclude that by replacing the quotient Π n/m  Π ‡ n/m by this quo- tient Q ∗∗∗ , we may assume without loss of generality that the quotient Π n/m  Π ‡ n/m induces the quotient Π n/n−1  Π ∗∗ n/n−1 of Π n/n−1 . Next, let us observe that if we regard Π n /Π n/n−1 as the “Π n−1 ” associated to some stable log curve of type (g, r), then: • If we apply the induction hypothesis to the almost pro-l quo- tient Π n/m /Π n/n−1  Π ‡ n/m /Π ‡ n/n−1 , then we obtain an [F- characteristic SA-maximal] almost pro-l quotient Π n /Π n/n−1  Q ‡ of Π n /Π n/n−1 which induces a quotient of Π n/m /Π n/n−1 that dominates the quotient Π n/m /Π n/n−1  Π ‡ n/m /Π ‡ n/n−1 . • If we apply the induction hypothesis to any almost pro-l quo- tient of Π n /Π n/n−1 that dominates both Q and Q ‡ [e.g., the quo- tient determined by the intersection of the kernels determined by the quotients Q, Q ‡ ], then we obtain an F-characteristic SA-maximal almost pro-l quotient Π n /Π n/n−1  (Π n /Π n/n−1 ) ∗∗ of Π n /Π n/n−1 that dominates Q and, moreover, induces a quo- tient of Π n/m /Π n/n−1 that dominates Π ‡ n/m /Π ‡ n/n−1 . In particu- lar, the above outer action Π n /Π n/n−1 → Out(Π ∗∗ n/n−1 ) factors through the natural surjection Π n /Π n/n−1  (Π n /Π n/n−1 ) ∗∗ . COMBINATORIAL ANABELIAN TOPICS III def 31 out ∗∗ ∗∗ Now let us write Π ∗∗ [cf. the discussion n = Π n/n−1  (Π n /Π n/n−1 ) entitled “Topological groups” in [CbTpI], §0 — where we note that Π ∗∗ n/n−1 is center-free by assertion (i)]. Then it follows immediately from Lemma 2.2 [which allows one to reduce an inclusion assertion concerning “Ker(−)’s” to an inclusion assertion concerning centraliz- ers] and the various definitions involved, together with our assumption that the quotient Π n/m  Π ‡ n/m induces the quotient Π n/n−1  Π ∗∗ n/n−1 of Π n/n−1 , that Π ∗∗ n is an SA-maximal almost pro-l quotient of Π n such that the quotient of Π n/m determined by Π n  Π ∗∗ n dominates the quo- ‡ tient Π n/m  Π n/m . Finally, it follows immediately from Lemma 2.2 [which allows one to reduce an F-characteristicity assertion concerning “Ker(−)” to an F-characteristicity assertion concerning a certain cen- tralizer], together with the fact that the quotients Π n/n−1  Π ∗∗ n/n−1 and Π n /Π n/n−1  (Π n /Π n/n−1 ) ∗∗ are F-characteristic, that Π ∗∗ n is F- characteristic. This completes the proof of assertion (ii). Assertion (iii) follows immediately from assertion (ii), together with Proposition 1.7, (viii). This completes the proof of Proposition 2.3.  def Definition 2.4. In the notation of Definition 2.1, write Π F = Π 2/1 , def def Π T = Π 2 , Π B = Π 1 ; thus, we have a natural exact sequence of profinite groups 1 −→ Π F −→ Π T −→ Π B −→ 1 [cf. the notation introduced in [CbTpI], Definition 6.3]. Let Π F  Π F ∗ be a maximal almost pro-Σ quotient of Π F [cf. Definition 1.1]. Then we shall say that Π F  Π ∗ F is base-admissible if the kernel of Π F  Π ∗ F is normal in Π T . Thus, if Π F  Π ∗ F is base-admissible, then the quotient Π F  Π ∗ F determines a quotient Π T  Π ∗ T of Π T which fits into a natural commutative diagram of profinite groups 1 −−−→ Π F −−−→ Π T −−−→ Π B −−−→ 1 ⏐ ⏐  ⏐ ⏐     1 −−−→ Π ∗ F −−−→ Π ∗ T −−−→ Π B −−−→ 1 — where the horizontal sequences are exact, and the vertical arrows are surjective. Definition 2.5. In the notation of Definition 2.4, suppose that Π F  Π ∗ F is base-admissible [cf. Definition 2.4]; thus, we have a quotient Π T  Π ∗ T 32 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI of Π T that fits into the commutative diagram of Definition 2.4. Let x ∈ X(k) be a k-valued point of the underlying scheme X of X log . (i) We shall write Π G x  Π ∗G x [cf. [CbTpI], Definition 6.3, (i)] for the maximal almost pro-Σ quotient of Π G x determined by the quotient Π F  Π ∗ F and the ∼ isomorphism Π F → Π G x fixed in [CbTpI], Definition 6.3, (i). [Here, we note that this quotient Π G x  Π ∗G x is independent ∼ of the choice of isomorphism Π F → Π G x in [CbTpI], Definition ∼ 6.3, (i).] Thus, the fixed isomorphism Π F → Π G x induces an ∼ isomorphism of profinite groups Π ∗ F → Π ∗G x . (ii) For c ∈ Cusp F (G) [cf. [CbTpI], Definition 6.5, (i)], we shall refer to a closed subgroup of Π ∗ F obtained by forming the im- ∼ age — via the isomorphism Π ∗G x ← Π ∗ F [cf. (i)] for some k- valued point x ∈ X(k) — of a cuspidal subgroup of Π ∗G x as- sociated to the cusp of G x corresponding to c ∈ Cusp F (G) [cf. [CbTpI], Lemma 6.4, (ii)] as a cuspidal subgroup of Π ∗ F associ- ated to c ∈ Cusp F (G). Note that it follows immediately from [CbTpI], Lemma 6.4, (ii), that the Π ∗ F -conjugacy class of a cus- pidal subgroup of Π ∗ F associated to c ∈ Cusp F (G) depends only on c ∈ Cusp F (G), i.e., it does not depend on the choice of x or on the choices of isomorphisms made in [CbTpI], Definition 6.3, (i). (iii) Recall that Π T = Π 2 , Π F = Π 2/1 [cf. Definition 2.4]. In par- ticular, it makes sense to speak of F-/C-/FC-admissible auto- morphisms or outomorphisms of Π ∗ T , Π ∗ F [cf. Definition 2.1, (v), (vi)]. Lemma 2.6 (Maximal almost pro-Σ quotients of VCN-sub- groups). In the notation of Definition 2.5, let Π ∗ c F ⊆ Π ∗ F be a cuspidal diag subgroup of Π ∗ F [cf. Definition 2.5, (ii)] associated to c Fdiag ∈ Cusp F (G) ∗ ⊆ Π ∗ F for the normal [cf. [CbTpI], Definition 6.5, (i)]. Write N diag ∗ closed subgroup of Π F topologically normally generated by Π ∗ c F ⊆ Π ∗ F . diag [Note that it follows immediately from [CbTpI], Lemma 6.4, (i), (ii), ∗ is normal in Π ∗ T .] Then the following hold: that N diag ∗ (i) If we regard Π ∗ F /N diag as a quotient of Π G by means of the ∼ natural outer isomorphism Π F /N diag → Π G of [CbTpI], Lemma ∗ , then 6.6, (i), and the natural surjection Π F /N diag  Π ∗ F /N diag ∗ ∗ Π F /N diag is a maximal almost pro-Σ quotient of Π G [cf. Definition 1.1]. COMBINATORIAL ANABELIAN TOPICS III 33 (ii) Let z F ∈ VCN(G x ), Π z F ⊆ Π G x a VCN-subgroup of Π G x asso- ciated to z F , and Π z F  Π z ‡ F an almost pro-Σ quotient of Π z F . Then there exists a base-admissible [cf. Definition 2.4] maximal almost pro-Σ quotient Π ∗∗ F of Π F such that the ∗∗ quotient Π z F  Π z F determined by the quotient Π F  Π ∗∗ F ‡ dominates the quotient Π z F  Π z F [cf. the discussion enti- tled “Topological groups” in §0]. (iii) Let z F ∈ VCN(G x ) \ {c Fdiag } and Π ∗ z F ⊆ Π ∗G x a VCN-subgroup of Π ∗G x associated to z F [cf. Definition 1.6, (i)]. Suppose that either • z F ∈ Edge(G x ) or • z F = v x F for v ∈ Vert(G) [cf. [CbTpI], Definition 6.3, (ii)] such that x does not lie on v [cf. [CbTpI], Definition 6.3, (iii)]. Then there exist a maximal almost pro-Σ quotient Π ∗∗ F of ∗∗ ∗∗ ⊆ Π of Π associated to z F Π F and a VCN-subgroup Π ∗∗ G x G x z F such that the following conditions are satisfied: ∗ (a) Π F  Π ∗∗ F dominates Π F  Π F . (b) Π F  Π ∗∗ F is base-admissible. (c) The quotient of Π ∗∗ z F determined by the composite ∼ ∗∗ ∗∗ ∗ Π ∗∗ z F → Π G x ← Π F  Π F factors through the quotient of Π ∗∗ z F determined by the com- posite ∼ ∗∗ ∗∗ ∗∗ ∗∗ Π ∗∗ z F → Π G x ← Π F  Π F /N diag ∗∗ — where we write N diag for the normal closed subgroup of ∗∗ Π F topologically normally generated by the cuspidal sub- F F groups of Π ∗∗ F associated to c diag ∈ Cusp (G). Proof. Assertion (i) follows immediately from Lemma 1.2, (i). Asser- tion (ii) follows immediately from Proposition 1.7, (viii), together with Lemma 1.2, (iii) [cf. also Proposition 1.7, (i)]. In a similar vein, as- sertion (iii) follows immediately, in light of the injectivity assertion of [CbTpI], Lemma 6.6, (iii), from Proposition 1.7, (viii) [applied to Π F /N diag ], together with Lemma 1.2, (iii) [cf. also Proposition 1.7, (i)]. This completes the proof of Lemma 2.6.  Lemma 2.7 (Outomorphisms that preserve the diagonal). In the notation of Lemma 2.6, let α  ∗ be an automorphism of Π ∗ T over Π B 34 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [i.e., which preserves and induces the identity automorphism on the quotient Π ∗ T  Π B ]. Write α F ∗ ∈ Out(Π ∗ F ) for the outomorphism of Π ∗ F determined by of α  ∗ . Then the following hold: (i) Suppose that α  ∗ preserves Π ∗ c F diag ⊆ Π ∗ F . Then the automor- ∗ induced by α  ∗ is the identity automor- phism of Π ∗ F /N diag phism. (ii) Let e ∈ Edge(G), x ∈ X(k) be such that x  e [cf. [CbTpI], Definition 6.3, (iii)]. Suppose that α F ∗ is C-admissible [cf. Definition 2.5, (iii)], and that Edge(G) = {e} ∪ Cusp(G). ∼ Then it holds that α F ∗ ∈ Out grph (Π ∗G x ) (⊆ Out(Π ∗G x ) ← Out(Π ∗ F )) [cf. Definition 1.6, (iii)]. If, moreover, α  ∗ preserves Π ∗ c F ⊆ Π ∗ F , then α F ∗ ∈ Out tion 1.8]. |grph| diag (Π ∗G x ) (⊆ Out grph (Π ∗G x )) [cf. Defini- (iii) If α  ∗ is FC-admissible [cf. Definition 2.5, (iii)], then α  ∗ ∗ ∗ ∗ preserves the Π F -conjugacy class of Π c F ⊆ Π F . diag def Proof. First, we verify assertion (i). Write D ∗ = N Π ∗ T (Π ∗ c F ) ⊆ Π ∗ T . diag Then it follows immediately from Proposition 1.7, (vii), that the nat- ural inclusion D ∗ → Π ∗ T fits into the following exact sequence 1 −−−→ Π ∗ c F −−−→ D ∗ −−−→ Π B −−−→ 1 ⏐ diag ⏐  ⏐ ⏐     1 −−−→ Π ∗ F −−−→ Π ∗ T −−−→ Π B −−−→ 1 — where the horizontal sequences are exact. Thus, assertion (i) follows immediately from a similar argument to the argument applied in the proof of the first assertion of [CbTpI], Lemma 6.7, (i) [cf. also the proof of [CmbCsp], Proposition 1.2, (iii)]. This completes the proof of assertion (i). Next, we verify assertion (ii). The fact that α F ∗ ∈ Out grph (Π ∗G x ) ∼ (⊆ Out(Π ∗G x ) ← Out(Π ∗ F )) follows immediately from Corollary 1.12, to- gether with a similar argument to the argument applied in the proof of the first assertion of [CbTpI], Lemma 6.7, (ii). Now suppose, moreover, that α  ∗ preserves Π ∗ c F ⊆ Π ∗ F . Then the fact that α F ∗ ∈ Out |grph| (Π ∗G x ) diag (⊆ Out grph (Π ∗G x )) follows immediately from assertion (i); Lemma 2.6, (i); Proposition 1.7, (iii), (v), together with a similar argument to the argument applied in the proof of the second assertion of [CbTpI], Lemma 6.7, (ii). This completes the proof of assertion (ii). Finally, assertion (iii) follows immediately, in light of Lemma 2.6, (i), from the definition of FC-admissibility [cf. also Proposition 1.7, (v)]. This completes the proof of Lemma 2.7.  COMBINATORIAL ANABELIAN TOPICS III 35 Lemma 2.8 (Triviality of certain outomorphisms). In the nota- tion of Definition 2.5, there exists a base-admissible maximal al- most pro-Σ quotient Π F  Π ∗∗ F [cf. Definitions 1.1; 2.4] of Π F that ∗ dominates Π F  Π F [cf. the discussion entitled “Topological groups” in §0] such that the following condition (‡) is satisfied: ∗ (‡): Let α  ∗ be an automorphism of Π T . Then for any base-admissible maximal almost pro-Σ quotient Π F  Π ∗∗∗ that dominates Π F  Π ∗∗  ∗ arises F F , if α from an FC-admissible automorphism [cf. Defini- tion 2.5, (iii)] of Π ∗∗∗ [where we write Π ∗∗∗ for the T T ∗ ∗∗∗ ∗∗∗ ∗∗∗ ∼ “Π T ” determined by Π F ] over Π T /Π F → Π B , then α  ∗ is Π ∗ F -inner. Proof. The following argument is essentially the same as the argument applied in [CmbCsp], [NodNon], [CbTpI] to prove [CmbCsp], Corollary 2.3, (ii); [NodNon], Corollary 5.3; [CbTpI], Lemma 6.8, respectively. Let us fix a cuspidal subgroup Π ∗ c F ⊆ Π ∗ F of Π ∗ F [cf. Definition 2.5, diag (ii)] associated to c Fdiag ∈ Cusp F (G) [cf. [CbTpI], Definition 6.5, (i)]. Let Π F  Π ∗∗ F be a base-admissible maximal almost pro-Σ quotient of Π F that dominates Π F  Π ∗ F ; Π F  Π ∗∗∗ a base-admissible max- F  ∗ an imal almost pro-Σ quotient of Π F that dominates Π F  Π ∗∗ F ; α automorphism of Π ∗ T that arises from an FC-admissible automorphism ∗∗∗ ∼ α  ∗∗∗ of Π ∗∗∗ over Π ∗∗∗ → Π B . Here, let us observe that one T T /Π F ∗ verifies easily that α  is an FC-admissible automorphism of Π ∗ T over ∼  ∗ . Π ∗ T /Π ∗ F → Π B . Write α F ∗ for the outomorphism of Π ∗ F determined by α ∗ ∗ ∗ Observe that since α F preserves the Π F -conjugacy class of Π c F ⊆ Π ∗ F diag [cf. Lemma 2.7, (iii)], we may assume without loss of generality — by  ∗∗∗ — that α  ∗ preserves replacing α  ∗∗∗ by a suitable Π ∗∗∗ F -conjugate of α Π ∗ c F ⊆ Π ∗ F , and hence [cf. Lemma 2.7, (i), (ii)] that diag ∗ (a) the automorphism of Π ∗ F /N diag induced by α  ∗ is the identity automorphism; (b) for e ∈ Edge(G), x ∈ X(k) such that x  e [cf. [CbTpI], Definition 6.3, (iii)], if Edge(G) = {e} ∪ Cusp(G), then α F ∗ ∈ ∼ Out |grph| (Π ∗G x ) (⊆ Out(Π ∗G x ) ← Out(Π ∗ F )) [cf. Definition 1.8]. Now we claim that the following assertion holds: Claim 2.8.A: Lemma 2.8 holds if (g, r) = (0, 3). Indeed, write c 1 , c 2 , c 3 ∈ Cusp(G) for the three distinct cusps of G; v ∈ Vert(G) for the unique vertex of G. For i ∈ {1, 2, 3}, let x i ∈ X(k) be such that x i  c i . Next, let us observe that since our assumption that (g, r) = (0, 3) implies that Node(G) = ∅, it follows immediately ∼ from (b) that, for i ∈ {1, 2, 3}, the outomorphism α F ∗ of Π ∗G xi ← Π ∗ F 36 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∼ is ∈ Out |grph| (Π ∗G xi ) (⊆ Out(Π ∗G xi ) ← Out(Π ∗ F )). Next, let us fix a ver- ∼ ticial subgroup Π ∗ v x F ⊆ Π ∗G x 2 ← Π ∗ F associated to v x F 2 ∈ Vert(G x 2 ) [cf. 2 [CbTpI], Definition 6.3, (ii)]. Then since α F ∗ ∈ Out |grph| (Π ∗G x 2 ), it fol- lows immediately from the [easily verified] surjectivity of the composite ∼ ∗ ∗ Π ∗ v x F → Π ∗G x 2 ← Π ∗ F  Π ∗ F /N diag that there exists an N diag -conjugate 2 β  ∗ of α  ∗ such that β  ∗ (Π ∗ F ) = Π ∗ F . Thus, it follows immediately from v x 2 v x 2 Lemma 2.6, (iii) — by replacing Π ∗∗ F by a suitable base-admissible max- imal almost pro-Σ quotient Π F  Π ∗∗ F [i.e., a quotient as in Lemma 2.6, (iii)] that dominates the original Π F  Π ∗∗ F and applying the conclusion ∗ ∗ ∗  “ β (Π v x F ) = Π v x F ”, together with the property (a) discussed above, in 2 2  ∗∗∗ ∈ Aut(Π ∗∗∗ the case where “ α ∗ ” is taken to be α T ) — that we may assume without loss of generality that (‡ 1 ): β  ∗ fixes and induces the identity automorphism ∼ on Π ∗ v x F ⊆ Π ∗G x 2 ← Π ∗ F . 2 Next, let Π ∗ c F 1 ⊆ Π ∗ F be a cuspidal subgroup of Π ∗ F associated to c F1 ∈ Cusp F (G) [cf. [CbTpI], Definition 6.5, (i)] that is contained in ∼ ∼ Π ∗ v x F ⊆ Π ∗G x 2 ← Π ∗ F ; Π ∗ v x F ⊆ Π ∗G x 3 ← Π ∗ F a verticial subgroup associated 2 3 to v x F 3 ∈ Vert(G x 3 ) that contains Π ∗ c F ⊆ Π ∗ F . Then it follows from the 1 inclusion Π ∗ F ⊆ Π ∗ F , together with (‡ ), that β  ∗ (Π ∗ F ) = Π ∗ F . Thus, v x 2 c 1 1 c 1 c 1 ∼ ∗ ∗ since the verticial subgroup Π v x F ⊆ Π G x 3 ← Π ∗ F is the unique verticial 3 ∼ subgroup of Π ∗G x 3 ← Π ∗ F associated to v x F 3 ∈ Vert(G x 3 ) that contains Π ∗ c F [cf. Proposition 1.7, (v), (vi)], it follows immediately from the 1 fact that α F ∗ ∈ Out |grph| (Π ∗G x 3 ) that β  ∗ (Π ∗ v x F ) = Π ∗ v x F . In particular, 3 3 it follows immediately from Lemma 2.6, (iii) — by replacing Π ∗∗ F by a suitable base-admissible maximal almost pro-Σ quotient Π F  Π ∗∗ F [i.e., a quotient as in Lemma 2.6, (iii)] that dominates the original ∗  ∗ ∗ Π F  Π ∗∗ F and applying the conclusion “ β (Π v x F ) = Π v x F ”, together 3 3 with the property (a) discussed above, in the case where “ α ∗ ” is taken to be α  ∗∗∗ ∈ Aut(Π ∗∗∗ T ) — that we may assume without loss of generality that (‡ 2 ): β  ∗ fixes and induces the identity automorphism ∼ on Π ∗ v x F ⊆ Π ∗G x 3 ← Π ∗ F . 3 On the other hand, since Π ∗ F is topologically generated by Π ∗ v x F ⊆ ∼ ∼ 2 Π ∗G x 2 ← Π ∗ F and Π ∗ v x F ⊆ Π ∗G x 3 ← Π ∗ F [cf. [CmbCsp], Lemma 1.13], (‡ 1 ) 3 and (‡ 2 ) imply that β  ∗ induces the identity automorphism on Π ∗ F . This completes the proof of Claim 2.8.A. Next, we claim that the following assertion holds: COMBINATORIAL ANABELIAN TOPICS III 37 Claim 2.8.B: Lemma 2.8 holds if (g, r) = (1, 1). Indeed, let us first observe that by working with 2-cuspidalizable de- generation structures [cf. [CbTpII], Definition 3.23, (i), (v)] that arise scheme-theoretically via a specialization isomorphism as in the discus- sion preceding [CmbCsp], Definition 2.1 [cf. also [CbTpI], Remark 5.6.1], we may switch back and forth, at will, between the case of smooth and non-smooth “X log ”. In particular, we may assume without loss of generality that (Vert(G)  , Cusp(G)  , Node(G)  ) = (1, 1, 1). Let v be the unique vertex of G, c the unique cusp of G, e the unique node of G, x ∈ X(k) such that x  c [cf. [CbTpI], Definition 6.3, (iii)], and H the sub-semi-graph of PSC-type [cf. [CbTpI], Definition 2.2, (i)] of the underlying semi-graph G x of G x whose set of vertices = {v x F } [cf. [CbTpI], Definition 6.3, (ii)]. Then it follows from [CbTpI], Lemma 6.4, (iv), that there exists a unique node e Fnew,x of G x such F ) [cf. [CbTpI], Lemma 6.4, (iii)]. Thus, one that e Fnew,x ∈ N (v new,x verifies easily that there exists a unique element e F x ∈ N (v x F ) such that N (v x F ) = {e Fnew,x , e F x }. Let us fix ∼ • a nodal subgroup Π ∗ e Fnew,x ⊆ Π ∗G x ← Π ∗ F associated to e Fnew,x [cf. Figure 1 below]. H ◦ • e Fnew,x F v new,x v x F • e F x ◦ Π ∗ sub F Figure 1: G x Then it follows immediately — by applying Proposition 1.7, (v), (vi), in the situation that arises in the case of a smooth “X log ” of type (1, 1) [cf. the observations made above concerning degeneration structures] — that there exist ∼ ⊆ Π ∗G x ← Π ∗ F associated to • a unique verticial subgroup Π ∗ v new,x F F v new,x and 38 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∼ • a unique subgroup Π ∗ (G x )| H ⊆ Π ∗G x ← Π ∗ F that belongs to the Π ∗ F -conjugacy class of subgroups that arises as the image of the natural outer homomorphism Π (G x )| H → Π G x  Π ∗G x [cf. [CbTpI], Definition 2.2, (ii)] such that Π ∗ e Fnew,x ⊆ Π ∗ v new,x , Π ∗ (G x )| H . Moreover, one verifies easily — F by applying the property (b) discussed above in the situation that arises in the case of a smooth “X log ” of type (1, 1) [cf. the observa- tions made above concerning degeneration structures] — that α F ∗ pre- ∼ serves the Π ∗ F -conjugacy classes of Π ∗ e Fnew,x , Π ∗ v new,x , Π ∗ (G x )| H ⊆ Π ∗G x ← Π ∗ F . F Thus, it follows immediately from the commensurable terminality of ∼ ∗ [cf. the image of the composite Π ∗ e Fnew,x → Π ∗G x ← Π ∗ F  Π ∗ F /N diag Proposition 1.7, (vii); Lemma 2.6, (i)], together with the property (a) ∗ -conjugate β  ∗ of α  ∗ such discussed above, that there exists an N diag that β  ∗ (Π ∗ e Fnew,x ) = Π ∗ e Fnew,x . In particular, in light of the uniqueness and Π ∗ (G x )| H , properties applied above to specify the subgroups Π ∗ v new,x F we conclude that β  ∗ (Π ∗ F ) = Π ∗ F , β  ∗ (Π ∗ ) = Π ∗ . Thus, v new,x v new,x (G x )| H (G x )| H it follows immediately from Lemma 2.6, (iii) — by replacing Π ∗∗ F by a suitable base-admissible maximal almost pro-Σ quotient Π F  Π ∗∗ F [i.e., a quotient as in Lemma 2.6, (iii), applied in the situation that arises in the case of a smooth “X log ” of type (1, 1) — cf. the observations made above concerning degeneration structures] that dominates the original ∗  ∗ ∗ Π F  Π ∗∗ F and applying the conclusion “ β (Π (G x )| H ) = Π (G x )| H ”, to- gether with the property (a) discussed above, in the case where “ α ∗ ” ∗∗∗ ∗∗∗ is taken to be α  ∈ Aut(Π T ) — that we may assume without loss of generality that (‡ 3 ): β  ∗ fixes and induces the identity automorphism ∼ on Π ∗ (G x )| H ⊆ Π ∗G x ← Π ∗ F . Next, let us write ∼ • Π ∗ v x F ⊆ Π ∗G x ← Π ∗ F for the unique [cf. Proposition 1.7, (v), (vi)] verticial subgroup associated to v x F [cf. [CbTpI], Definition 6.3, (ii)] such that Π ∗ e Fnew,x ⊆ Π ∗ v x F ⊆ Π ∗ (G x )| H . [Note that it follows immediately from the various definitions involved that such a verticial subgroup associated to v x F always exists.] Then since Π ∗ v x F ⊆ Π ∗ (G x )| H , it fol- lows from (‡ 3 ) that β  ∗ fixes and induces the identity automorphism ∼ [cf. the discus- on Π ∗ F ⊆ Π ∗ ← Π ∗ . Thus, since β  ∗ (Π ∗ F ) = Π ∗ F v x G x F v new,x v new,x sion preceding (‡ 3 )], we conclude that β  ∗ preserves the closed subgroup COMBINATORIAL ANABELIAN TOPICS III 39 Π ∗ F sub ⊆ Π ∗ F of Π ∗ F obtained by forming the image of the natural homo- morphism   lim Π ∗ v new,x ← Π ∗ e Fnew,x → Π ∗ v x F −→ Π ∗ F F −→ — where the inductive limit is taken in the category of profinite groups. Next, let us observe that the Π ∗ F -conjugacy class of Π ∗ F sub ⊆ Π ∗ F coin- cides with the Π ∗ F -conjugacy class of the image Π ∗ (G x ) F [cf. [CbTpI], {e x } Definition 2.5, (ii)] of the composite ∼ Π (G x ) {e F } → Π G x ← Π F  Π ∗ F x — where the first arrow is the natural outer injection discussed in [CbTpI], Proposition 2.11, and we recall that e F x is the node of G x that corresponds to the node e of G. On the other hand, if we write for the maximal almost pro-Σ quotient of Π (G x ) {e F } [cf. Π ∗ (G x ) F {e new } new [CbTpI], Definition 2.8] determined by the maximal almost pro-Σ quo- ∼ tient Π ∗G x and the natural outer isomorphism Φ (G x ) {e F } : Π (G x ) {e F } → new new Π G x [cf. [CbTpI], Definition 2.10], then Π ∗ F sub may be regarded as a ver- ∼ ∼ ticial subgroup of Π ∗ (G x ) F → Π ∗G x ← Π ∗ F [cf. [CbTpI], Proposition {e new } 2.9, (i), (3)]. Thus, it follows from Proposition 1.7, (vii), that Π ∗ F sub is commensurably terminal in Π ∗ F . Next, let us observe that, by applying a similar argument to the ar- gument given in [CmbCsp], Definition 2.1, (iii), (vi), or [NodNon], Def- inition 5.1, (ix), (x) [i.e., roughly speaking, by considering the portion of the underlying scheme X 2 of X 2 log corresponding to the underlying scheme (X v ) 2 of the 2-nd log configuration space (X v ) log 2 of the stable log log curve X v determined by G| v — cf. [CbTpI], Definition 2.1, (iii)], ∼ one concludes that there exists a verticial subgroup Π v ⊆ Π G ← Π B associated to v ∈ Vert(G) such that the outer action of Π v on Π ∗ F de- ρ ∗ 2/1 termined by the composite Π v → Π B → Out(Π ∗ F ) — where we write ρ ∗ 2/1 for the outer action determined by the exact sequence of profinite groups 1 −→ Π ∗ F −→ Π ∗ T −→ Π B −→ 1 — preserves the Π ∗ F -conjugacy class of the commensurably terminal subgroup Π ∗ F sub ⊆ Π ∗ F [so we obtain a natural outer representation Π v → Out(Π ∗ F sub ) — cf. [CbTpI], Lemma 2.12, (iii)], and, moreover, def out that if we write Π ∗ T sub = Π ∗ F sub  Π v (⊆ Π ∗ T ) [cf. the discussion enti- tled “Topological groups” in [CbTpI], §0], then it follows from Propo- sition 1.7, (ii), that Π ∗ T sub is naturally isomorphic to a profinite group of the form “Π ∗ T ” obtained by taking “G” to be G| v . Now since β  ∗ (Π ∗ F sub ) = Π ∗ F sub , and α  ∗ is an automorphism over the ∼ quotient Π ∗ F /Π ∗ T → Π B , one verifies immediately that β  ∗ determines an ∗ automorphism β  T ∗ sub of Π T sub over Π v . Thus, since G| v is of type (0, 3) 40 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [cf. [CbTpI], Definition 2.3, (i)], by considering a diagram similar to the diagram of [CmbCsp], Definition 2.1, (vi), or [NodNon], Definition 5.1, (x), and applying Claim 2.8.A [cf. also Lemma 2.6, (ii)], we conclude — by replacing Π ∗∗ F by a suitable base-admissible maximal almost pro-Σ ∗∗ quotient Π F  Π ∗∗ F that dominates the original Π F  Π F and applying the conclusion that β  ∗ determines an automorphism of Π ∗ T sub over Π v in the case where “ α ∗ ” is taken to be α  ∗∗∗ ∈ Aut(Π ∗∗∗ T ) — that we may assume without loss of generality that (‡ 4 ): β  T ∗ sub is a Π ∗ F sub -inner automorphism. Moreover, since β  ∗ fixes and induces the identity automorphism on Π ∗ v x F [cf. the discussion following (‡ 3 )], and Π ∗ v x F is commensurably ter- minal in [Π ∗ F , hence also in] Π ∗ F sub [cf. Proposition 1.7, (vii)] and slim [cf. Proposition 1.7, (ii)], we conclude that β  T ∗ sub is the identity auto- morphism; in particular, since Π ∗ v new,x ⊆ Π ∗ F sub , β  ∗ induces the identity F . Thus, since Π ∗ F is topologically generated by automorphism on Π ∗ v new,x F [cf. [CmbCsp], Proposition 2.2, (iii)], it follows from Π ∗ (G x )| H and Π ∗ v new,x F (‡ 3 ) that β  ∗ is the identity automorphism. This completes the proof of Claim 2.8.B. Finally, we claim that the following assertion holds: Claim 2.8.C: Lemma 2.8 holds for arbitrary (g, r). We verify Claim 2.8.C by induction on 3g − 3 + r. If 3g − 3 + r = 0, i.e., (g, r) = (0, 3), then Claim 2.8.C amounts to Claim 2.8.A. On the other hand, if (g, r) = (1, 1), then Claim 2.8.C amounts to Claim 2.8.B. Thus, we suppose that 3g − 3 + r > 0, that (g, r)  = (1, 1), and that the induction hypothesis is in force. Since 3g − 3 + r > 0 and (g, r)  = (1, 1), one verifies easily that there exists a stable log curve Y log of type (g, r) over (Spec k) log such that Y log has precisely one node and precisely two vertices. Thus, by working with 2-cuspidalizable de- generation structures [cf. [CbTpII], Definition 3.23, (i), (v)] that arise scheme-theoretically via a specialization isomorphism as in the discus- sion preceding [CmbCsp], Definition 2.1 [cf. also [CbTpI], Remark 5.6.1], we may replace X log by Y log and assume without loss of gener- ality that (Vert(G)  , Node(G)  ) = (2, 1). Let e be the unique node of G and x ∈ X(k) such that x  e [cf. [CbTpI], Definition 6.3, (iii)]. Next, let us observe that since Node(G)  = {e}  = 1, it follows from the property (b) discussed above ∼ that α F ∗ ∈ Out |grph| (Π ∗G x ) (⊆ Out(Π ∗G x ) ← Out(Π ∗ F )). Write {e F1 , e F2 } = F N (v new,x ) [cf. [CbTpI], Lemma 6.4, (iv)]. Also, for i ∈ {1, 2}, denote by v i ∈ Vert(G) the vertex of G such that (v i ) F x ∈ Vert(G x ) [cf. [CbTpI], F Definition 6.3, (ii)] is the unique element of V(e F i )\{v new,x } [cf. [CbTpI], Lemma 6.4, (iv)]; by H i the sub-semi-graph of PSC-type [cf. [CbTpI], COMBINATORIAL ANABELIAN TOPICS III 41 Definition 2.2, (i)] of the underlying semi-graph G x of G x whose set of F , (v i ) F x } [cf. Figure 2 below]. vertices = {v new,x ∼ For i ∈ {1, 2}, let Π ∗ (v i ) F x ⊆ Π ∗G x ← Π ∗ F be a verticial subgroup of ∼ F }. Then since Π ∗G x ← Π ∗ F associated to the vertex (v i ) F x ∈ V(e F i ) \ {v new,x |grph| ∗ ∗ ∗ ∗ (Π G x ), it follows that α  preserves the Π F -conjugacy class α F ∈ Out ∼ ∗ ∗ ∗ of Π (v i ) F x ⊆ Π G x ← Π F . Thus, since the image of the composite H 1 ◦ · · · · · · · ◦ F v new,x (v 1 ) F x • • e F1 ◦ (v 2 ) F x e F2 • ◦ · · · · · · · ◦ H 2 Figure 2: G x ∗ Π ∗ (v i ) F x → Π ∗ F  Π ∗ F /N diag is commensurably terminal [cf. Proposi- tion 1.7, (vii); Lemma 2.6, (i)], it follows immediately from the property ∗ (a) discussed above that there exists an N diag -conjugate β  i ∗ [which may depend on i ∈ {1, 2}!] of α  ∗ such that β  i ∗ (Π ∗ (v i ) F x ) = Π ∗ (v i ) F x . Therefore, it follows immediately from Lemma 2.6, (iii) — by replacing Π ∗∗ F by a suitable base-admissible maximal almost pro-Σ quotient Π F  Π ∗∗ F [i.e., a quotient as in Lemma 2.6, (iii)] that dominates the original Π F  Π ∗∗ F and applying the conclusion “ β  i ∗ (Π ∗ (v i ) F x ) = Π ∗ (v i ) F x ”, together with the property (a) discussed above, in the case where “ α ∗ ” is taken to be α  ∗∗∗ ∈ Aut(Π ∗∗∗ T ) — that we may assume without loss of generality that (‡ 5 ): β  i ∗ induces the identity automorphism of Π ∗ (v i ) F x . ∼ Next, let Π ∗ e F ⊆ Π ∗ (v i ) F x be a nodal subgroup of Π ∗G x ← Π ∗ F associated i ∼ ∗ ∗ to e F i ∈ Node(G x ) that is contained in Π ∗ (v i ) F x ; Π ∗ v new,x F ;i ⊆ Π G x ← Π F a verticial subgroup [which may depend on i ∈ {1, 2}!] associated to F v new,x ∈ Vert(G x ) that contains Π ∗ e F : i Π ∗ v new,x F ;i ⊇ Π ∗ e F i ⊆ Π ∗ (v i ) F x ⊆ ∼ Π ∗G x ← Π ∗ F . 42 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Then it follows from the inclusion Π ∗ e F ⊆ Π ∗ (v i ) F x , together with (‡ 5 ), i that β  ∗ (Π ∗ F ) = Π ∗ F . Since, moreover, Π ∗ F is the unique verti- i v new,x ;i e i ∼ ∗ ∗ F cial subgroup of Π G x ← Π F associated to v new,x that contains Π ∗ e F [cf. e i i Proposition 1.7, (v), (vi)], it follows immediately from the fact that ∗  ∗ α F ∗ ∈ Out |grph| (Π ∗G x ) that β  i ∗ (Π ∗ v new,x F F ;i ) = Π v new,x ;i . Thus, β i preserves the closed subgroup Π ∗ F i ⊆ Π ∗ F of Π ∗ F obtained by forming the image of the natural homomorphism   ∗ ∗ lim Π ∗ v new,x ← Π → Π −→ Π ∗ F F F F (v i ) x ;i e i −→ — where the inductive limit is taken in the category of profinite groups. Next, let us observe that the Π ∗ F -conjugacy class of Π ∗ F i ⊆ Π ∗ F coin- cides with the Π ∗ F -conjugacy class of the image Π ∗ (G x )| H [cf. [CbTpI], i Definition 2.2, (ii)] of the composite ∼ Π (G x )| H i → Π G x ← Π F  Π ∗ F — where the first arrow is the natural outer injection discussed in [CbTpI], Proposition 2.11. On the other hand, if we write Π ∗ (G x ) F for {e } i the maximal almost pro-Σ quotient of Π (G x ) {e F } [cf. [CbTpI], Definition i 2.8] determined by the maximal almost pro-Σ quotient Π ∗G x and the ∼ natural outer isomorphism Φ (G x ) {e F } : Π (G x ) {e F } → Π G x [cf. [CbTpI], i i Definition 2.10], then Π ∗ F i may be regarded as a verticial subgroup of ∼ ∼ Π ∗ (G x ) F → Π ∗G x ← Π ∗ F [cf. [CbTpI], Proposition 2.9, (i), (3)]. Thus, it {e } i follows from Proposition 1.7, (vii), that Π ∗ F i is commensurably terminal in Π ∗ F . Moreover, by applying a similar argument to the argument given in [CmbCsp], Definition 2.1, (iii), (vi), or [NodNon], Definition 5.1, (ix), (x) [i.e., roughly speaking, by considering the portion of the underlying scheme X 2 of X 2 log corresponding to the underlying scheme (X v i ) 2 of log the 2-nd log configuration space (X v i ) log 2 of the stable log curve X v i determined by G| v i — cf. [CbTpI], Definition 2.1, (iii)], one concludes ∼ that there exists a verticial subgroup Π v i ⊆ Π G ← Π B associated to v i ∈ Vert(G) such that the outer action of Π v i on Π ∗ F determined by ρ ∗ 2/1 the composite Π v i → Π B → Out(Π ∗ F ) — where we write ρ ∗ 2/1 for the outer action determined by the exact sequence of profinite groups 1 −→ Π ∗ F −→ Π ∗ T −→ Π B −→ 1 — preserves the Π ∗ F -conjugacy class of the commensurably terminal subgroup Π ∗ F i ⊆ Π ∗ F [so we obtain a natural outer representation Π v i → Out(Π ∗ F i ) — cf. [CbTpI], Lemma 2.12, (iii)], and, moreover, that if we def out write Π ∗ T i = Π ∗ F i  Π v i (⊆ Π ∗ T ) [cf. the discussion entitled “Topolog- ical groups” in [CbTpI], §0], then it follows from Proposition 1.7, (ii), COMBINATORIAL ANABELIAN TOPICS III 43 that Π ∗ T i is naturally isomorphic to a profinite group of the form “Π ∗ T ” obtained by taking “G” to be G| v i .  ∗ is an automorphism over the quo- Now since β  i ∗ (Π ∗ F i ) = Π ∗ F i , and α ∼ ∗ → Π B , one verifies immediately that β  i determines an au- tient Π ∗ F /Π T tomorphism β  T ∗ i of Π ∗ T i over Π v i . Thus, since the quantity “3g − 3 + r” associated to G| v i is < 3g−3+r, by considering a diagram similar to the diagram of [CmbCsp], Definition 2.1, (vi), or [NodNon], Definition 5.1, (x), and applying the induction hypothesis [cf. also Lemma 2.6, (ii)], we conclude — by replacing Π ∗∗ F by a suitable base-admissible maximal ∗∗ almost pro-Σ quotient Π F  Π ∗∗ F that dominates the original Π F  Π F and applying the conclusion that β  i ∗ determines an automorphism of α ∗ ” is taken to be α  ∗∗∗ ∈ Aut(Π ∗∗∗ Π ∗ T i over Π v i in the case where “ T ) — that we may assume without loss of generality that (‡ ): β  ∗ is a Π ∗ -inner automorphism. 6 T i F i In particular, it follows immediately, by allowing i ∈ {1, 2} to vary, from Proposition 1.7, (vii) [which implies the commensurable terminal- ity of Π ∗ (v i ) F x ⊆ Π F ∗ i ], that the outomorphisms of Π ∗ (G x ) F , Π ∗ (G x ) F {e 1 } obtained by conjugating α F ∗ by the isomorphisms Π ∗ (G x ) [induced by Φ (G x ) {e F } ], Π ∗ (G x ) 1 ∼ {e F 2 } profinite Dehn multi-twists of Π ∗ (G x ) {e F 1 } ∼ {e 2 } → Π ∗G x → Π ∗G x [induced by Φ (G x ) {e F } ] are , Π ∗ (G x ) F {e 1 } 2 , respectively. Thus, F {e 2 } it follows from Lemma 1.10 that α F ∗ is the identity outomorphism. This completes the proof of Claim 2.8.C, hence also of Lemma 2.8.  Theorem 2.9 (Almost pro-Σ analogue of the injectivity por- tion of the theory of combinatorial cuspidalization). Let Σ be a nonempty set of prime numbers, n a positive integer, (g, r) a pair of nonnegative integers such that 2g − 2 + r > 0, and X a hyperbolic curve of type (g, r) over an algebraically closed field of characteristic zero. For each positive integer i, write X i for the i-th configuration space of X [cf. [MzTa], Definition 2.1, (i)]; Π i for the pro-Primes configuration space group [cf. [MzTa], Definition 2.3, (i)] given by the étale fundamental group π 1 (X i ) of X i . Also, we shall write pr : X n+1  X n for the projection obtained by forgetting the (n + 1)- st factor and Π n+1/n ⊆ Π n+1 for the kernel of some fixed surjection pr Π : Π n+1  Π n [that belongs to the collection of surjections that con- stitutes the outer surjection] induced by pr. Let Π n+1  Π ∗ n+1 be a quotient of Π n+1 such that the quotient Π ∗ n+1/n of Π n+1/n ⊆ Π n+1 de- termined by the quotient Π n+1  Π ∗ n+1 is a maximal almost pro-Σ quotient of Π n+1/n [cf. Definition 1.1]. Then there exists a quotient Π n+1  Π ∗∗ n+1 of Π n+1 such that the following conditions are satisfied: 44 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (i) The quotient Π n+1  Π ∗∗ n+1 dominates [cf. the discussion entitled “Topological groups” in §0] the quotient Π n+1  Π ∗ n+1 ∗ [i.e., Π n+1  Π ∗∗ n+1  Π n+1 ]. (ii) The quotient Π ∗∗ n+1/n of Π n+1/n ⊆ Π n+1 determined by the quo- tient Π n+1  Π ∗∗ n+1 is a maximal almost pro-Σ quotient of Π n+1/n . (iii) Let α ∗ be an outomorphism of Π ∗ n+1 and Π n+1  Π ∗∗∗ n+1 a ∗∗ quotient that dominates the quotient Π n+1  Π n+1 and in- duces a maximal almost pro-Σ quotient Π ∗∗∗ n+1/n of Π n+1/n . ∗ Suppose that α arises from an FC-admissible [cf. Defi- ∗∗∗ nition 2.1, (v)] automorphism α  ∗∗∗ of Π ∗∗∗ [i.e., n+1 over Π n ∗∗∗ which induces the identity automorphism of Π n ] — where we write Π ∗∗∗ for the quotient of Π n determined by the quotient n ∗ Π n+1  Π ∗∗∗ n+1 . Then α is the identity outomorphism. Proof. First, we claim that the following assertion holds: Claim 2.9.A: To verify Theorem 2.9, it suffices to verify Theorem 2.9 in the case where the kernel of the natural surjection Π n+1  Π ∗ n+1 is contained in Π n+1/n , i.e., the natural surjection ∼ Π n ← Π n+1 /Π n+1/n  Π ∗ n+1 /Π ∗ n+1/n — where the first arrow is the natural isomorphism — is an isomorphism. Indeed, Claim 2.9.A follows immediately, by considering the objects obtained by base-changing the various objects that appear in the case ∼ of an arbitrary quotient Π n+1  Π ∗ n+1 via the natural surjection Π n ← Π n+1 /Π n+1/n  Π ∗ n+1 /Π ∗ n+1/n . By Claim 2.9.A, we may assume with- out loss of generality that the kernel of Π n+1  Π ∗ n+1 is contained in Π n+1/n . Next, we claim that the following assertion holds: Claim 2.9.B: To verify Theorem 2.9, it suffices to verify Theorem 2.9 in the case where n = 1. Indeed, suppose that n ≥ 2, and that Theorem 2.9 holds whenever n = 1. Write Π n+1/n−1 ⊆ Π n+1 for the kernel of the outer surjection Π n+1  Π n−1 induced by the projection X n+1 → X n−1 obtained by forgetting the (n + 1)-st and n-th factors of X n+1 ; Π n/n−1 ⊆ Π n for the kernel of the outer surjection Π n  Π n−1 induced by the projection X n → X n−1 obtained by forgetting the n-th factor of X n ; Π ∗ n+1/n−1 for ∗ the quotient of Π n+1/n−1 determined by the quotient Π n+1  Π n+1 . Then let us recall [cf. [MzTa], Proposition 2.4, (i)] that one may in- terpret the surjection Π ∗ n+1/n−1  Π n/n−1 induced by the fixed surjec- tion pr Π : Π n+1  Π n as the surjection “pr Π : Π ∗ 2  Π 1 ” in the case where “X” is of type (g, r + n − 1). Thus, by applying Theorem 2.9 COMBINATORIAL ANABELIAN TOPICS III 45 in the case where n = 1 to the quotient Π n+1/n−1  Π ∗ n+1/n−1 , we obtain a quotient Π ∗∗ n+1/n−1 of Π n+1/n−1 which satisfies conditions (i), (ii), (iii) in the statement of Theorem 2.9. [Here, we note that since the kernel of Π n+1/n−1  Π ∗ n+1/n−1 is contained in Π n+1/n , the kernel of Π n+1/n−1  Π ∗∗ n+1/n−1 is also contained in Π n+1/n .] Next, let N ⊆ Π n+1/n be a normal open subgroup of Π n+1/n with respect to which Π ∗∗ n+1/n is a maximal almost pro-Σ quotient of Π n+1/n . Then it follows immediately from Lemma 1.2, (iii) [cf. also [MzTa], Proposition 2.2, (ii)], that we may assume without loss of generality — by replacing N by a suitable normal open subgroup contained in N — ∗∗ that the kernel of Π n+1/n  Π ∗∗ n+1/n is normal in Π n+1 . Write Π n+1 for the quotient of Π n+1 by the kernel of Π n+1/n  Π ∗∗ n+1/n . Then it is im- ∗∗ mediate that this quotient Π n+1 satisfies conditions (i), (ii) in the state- ment of Theorem 2.9, and, moreover, that the kernel of Π n+1  Π ∗∗ n+1 is satisfies condition (iii) in the contained in Π n+1/n . To verify that Π ∗∗ n+1 ∗∗∗ statement of Theorem 2.9, let Π n+1  Π n+1 be a quotient as in condi- tion (iii) in the statement of Theorem 2.9 and α  ∗ an automorphism of  ∗∗∗ of Π ∗∗∗ Π ∗ n+1 which arises from an FC-admissible automorphism α n+1 ∗∗∗ over Π n . Then since α  is FC-admissible, it is immediate that α  ∗∗∗ ∗∗∗ ∗∗∗ preserves Π ∗∗∗ n+1/n−1 ⊆ Π n+1 , where we write Π n+1/n−1 for the quotient of Π n+1/n−1 determined by the quotient Π n+1  Π ∗∗∗ n+1 . In particular, it , together with the fact that α  ∗∗∗ is follows from our choice of Π ∗∗ n+1/n−1  ∗ is an automor- an automorphism of Π ∗∗∗ n+1 over Π n [which implies that α ∗ phism of Π n+1 over Π n ], that we may assume without loss of generality — i.e., by replacing α  ∗ by a suitable Π ∗ n+1/n−1 -conjugate, which may in fact [in light of the slimness of Π n/n−1 — cf., e.g., [CmbGC], Remark 1.1.3] be taken to be a Π ∗ n+1/n -conjugate — that the automorphism of  ∗ is the identity automorphism. Thus, since α  ∗ is an Π ∗ n+1/n induced by α automorphism of Π ∗ n+1 over Π n , and Π ∗ n+1/n is slim [cf. Proposition 1.7, ∼ out (i)], we may apply the natural isomorphism Π ∗ n+1 → Π ∗ n+1/n  Π n [cf. the discussion entitled “Topological groups” in [CbTpI], §0] to conclude [cf., e.g., [Hsh], Lemma 4.10] that the automorphism α  ∗ of Π ∗ n+1 is the identity automorphism. In particular, we conclude that Π ∗∗ n+1 satisfies condition (iii) in the statement of Theorem 2.9. This completes the proof of Claim 2.9.B. By Claim 2.9.B, we may assume without loss of generality that n = 1. On the other hand, if n = 1, then one verifies easily that Theorem 2.9 follows immediately from Lemma 2.8. This completes the proof of Theorem 2.9.  Corollary 2.10 (Almost pro-l analogue of the injectivity por- tion of the theory of combinatorial cuspidalization). Let l be a 46 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI prime number, n a positive integer, (g, r) a pair of nonnegative in- tegers such that 2g − 2 + r > 0, and X a hyperbolic curve of type (g, r) over an algebraically closed field of characteristic zero. For each positive integer i, write X i for the i-th configuration space of X [cf. [MzTa], Definition 2.1, (i)]; Π i for the pro-Primes configu- ration space group [cf. [MzTa], Definition 2.3, (i)] given by the étale fundamental group π 1 (X i ) of X i . Let Π n+1  Π ∗ n+1 be an F- characteristic SA-maximal almost pro-l quotient of Π n+1 [cf. Definition 2.1, (ii), (iii)]. Then there exists an F-characteristic SA- maximal almost pro-l quotient Π n+1  Π ∗∗ n+1 of Π n+1 such that Π n+1  Π ∗∗ dominates [cf. the discussion entitled “Topological n+1 ∗ groups” in §0] the quotient Π n+1  Π n+1 , and, moreover, satisfies the following property: For any F-characteristic SA-maximal almost pro-l quotient Π n+1  Π ∗∗∗ n+1 of Π n+1 that dominates the quotient , the image of the composite Π n+1  Π ∗∗ n+1   FC FC ∗ ∗∗∗ ∗∗∗  Π ) ∩ Ker Out (Π ) → Out (Π ) Out FC (Π ∗∗∗ n+1 n+1 n+1 n FC FC ∗ ∗ ∗∗∗ ∗ → Out FC (Π ∗∗∗ n+1  Π n+1 )  Out (Π n+1  Π n+1 ) → Out (Π n+1 ) [cf. Definition 2.1, (vii), (viii)] — where we write Π ∗∗∗ n for the quotient , and the homomor- of Π n determined by the quotient Π n+1  Π ∗∗∗ n+1 FC FC ∗∗∗ ∗∗∗ phism Out (Π n+1 ) → Out (Π n ) [in large parentheses] is the homo- morphism induced by the projection X n+1 → X n obtained by forgetting the (n + 1)-st factor — is trivial. Proof. This follows immediately from Theorem 2.9, together with Propo- sition 2.3, (ii).  Remark 2.10.1. (i) Theorem 2.9 and Corollary 2.10 may be regarded, respectively, as almost pro-Σ, almost pro-l versions of the injectivity portion of [NodNon], Theorem B. In this context, it is of interest to recall that the pro-l version of this sort of injectivity result may also be obtained by means of the Lie-theoretic approach of [Tk]. On the other hand, it does not appear, at the time of writing, that this Lie-theoretic approach may be extended so as to yield an alternate proof either of the profinite portion of the injectivity result of [NodNon], Theorem B, or of the almost pro-Σ/pro-l versions of this result given in Theorem 2.9, Corollary 2.10 of the present paper. (ii) In the context of the observations of (i), it is of interest to recall that the various injectivity results of [NodNon] and the present paper that are discussed in (i) are obtained as consequences of COMBINATORIAL ANABELIAN TOPICS III 47 various combinatorial versions of the Grothendieck Conjecture. From this point of view, it seems natural to pose the following question: Is it possible to prove a Lie-theoretic combinatorial version of the Grothendieck Conjecture that allows one to derive the Lie-theoretic injectivity results of [Tk] by means of techniques analogous to the tech- niques applied in [NodNon] and the present paper? At the time of writing, it is not clear to the authors whether or not this question may be answered in the affirmative. In the remainder of §2, we consider an almost pro-l analogue of the tripod homomorphism of [CbTpII], Definition 3.19. Here, we recall that, as discussed in [CbTpII], Remark 3.19.1, the tripod homomorphism may be understood as a sort of abstract combinatorial analogue of the natural surjection from the arithmetric fundamental group of a moduli stack of curves over an arithmetic base field to the absolute Galois group of the base field. Lemma 2.11 (Commensurators of various subgroups of geo- metric origin). We shall apply the notational conventions established in §3 of [CbTpII]. In the notation of [CbTpII], Lemma 3.6, sup- pose that (j, i) = (1, 2); E = {i, j}; z i,j,x ∈ Edge(G j∈E\{i},x ). [Thus, ∼ G j∈E\{i},x = G i∈E\{j},x = G; Π 2 = Π E ; Π 1 = Π {j} → Π G j∈E\{i},x = Π G ; ∼ def def Π 2/1 = Π E/(E\{i}) → Π G i∈E,x .] Write G 2/1 = G i∈E,x ; G 1\2 = G j∈E,x ; def def ∼ Π Π p Π 1\2 = p E/{2} : Π 2  Π {2} ; Π 1\2 = Ker(p 1\2 ) = Π E/{2} → Π G 1\2 ; def def z x = z i,j,x ∈ Edge(G); c diag = c diag i,j,x ∈ Cusp(G 2/1 ) [cf. the notation def new ∈ Vert(G 2/1 ) [cf. the no- of [CbTpII], Lemma 3.6, (ii)]; v new = v i,j,x tation of [CbTpII], Lemma 3.6, (iv)]. Let Π z x ⊆ Π 1 be an edge-like subgroup associated to z x ∈ Edge(G); Π v new ⊆ Π 2/1 a verticial sub- group associated to v new ; Π c diag ⊆ Π 2/1 a cuspidal subgroup associated to c diag that is contained in Π v new [cf. [CbTpII], Lemma 3.6, (iv)]. Let Π 2  Π ∗ 2 be an SA-maximal almost pro-l quotient of Π 2 [cf. Definition 2.1, (ii)]. Write Π ∗ 2/1 , Π ∗ 1\2 , Π ∗ 1 , Π ∗{2} for the respective quo- tients of Π 2/1 , Π 1\2 , Π 1 , Π {2} determined by the quotient Π 2  Π ∗ 2 of Π 2 ; Π ∗G , Π ∗G 2/1 for the respective quotients of Π G , Π G 2/1 determined by ∼ the quotients Π 1  Π ∗ 1 , Π 2/1  Π ∗ 2/1 and the isomorphisms Π 1 → Π G , ∼ ∗ ∗ ∗ Π 2/1 → Π G 2/1 fixed in [CbTpII], Definition 3.1, (iii); (p Π 2/1 ) : Π 2  Π 1 , ∗ ∗ ∗ (p Π 1\2 ) : Π 2  Π {2} for the respective natural surjections induced by Π ∗ ∗ ∗ ∗ ∗ p Π 2/1 : Π 2  Π 1 , p 1\2 : Π 2  Π {2} ; Π z x ⊆ Π 1 , Π c diag ⊆ Π v new ⊆ Π 2/1 for the respective images of Π z x ⊆ Π 1 , Π c diag ⊆ Π v new ⊆ Π 2/1 in Π ∗ 1 , 48 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI def def def Π ∗ 2/1 ; Π ∗ 2 | z x = Π ∗ 2 × Π ∗ 1 Π ∗ z x ⊆ Π ∗ 2 ; D c ∗ diag = N Π ∗ 2 (Π ∗ c diag ); I v ∗ new | z x = def Z Π ∗ 2 | zx (Π ∗ v new ) ⊆ D v ∗ new | z x = N Π ∗ 2 | zx (Π ∗ v new ). Then the following hold: (i) It holds that D c ∗ diag ∩ Π ∗ 2/1 = C Π ∗ 2 (Π ∗ c diag ) ∩ Π ∗ 2/1 = Π ∗ c diag . (ii) It holds that C Π ∗ 2 (Π ∗ c diag ) = D c ∗ diag . ∗ ∗ ∗ (iii) The surjection (p Π 2/1 ) : Π 2  Π 1 determines an isomorphism ∼ D c ∗ diag /Π ∗ c diag → Π ∗ 1 . Moreover, the composite ∼ Π 1  Π 1 ∗ ← D c ∗ diag /Π ∗ c diag  Π ∗{2} — where the first arrow is the natural surjection, the second arrow is the isomorphism obtained above, and the third arrow ∗ ∗ ∗ is the surjection determined by (p Π 1\2 ) : Π 2  Π {2} — coin- cides, up to composition with an inner automorphism, with the natural surjection Π 1  Π ∗{2} . (iv) The composite I v ∗ new | z x → D v ∗ new | z x → Π ∗ z x is an isomorphism. (v) The natural inclusions Π ∗ v new , I v ∗ new | z x → D v ∗ new | z x determine an ∼ isomorphism Π ∗ v new × I v ∗ new | z x → D v ∗ new | z x = C Π ∗ 2 | zx (Π ∗ v new ). (vi) It holds that C Π ∗ 2 (D v ∗ new | z x ) ⊆ C Π ∗ 2 (Π ∗ v new ). (vii) D v ∗ new | z x is commensurably terminal in Π ∗ 2 . Proof. First, we verify assertion (i). Observe that we have inclusions Π ∗ c diag ⊆ D c ∗ diag ⊆ C Π ∗ 2 (Π ∗ c diag ). Thus, since Π c ∗ diag is commensurably ter- minal in Π ∗ 2/1 [cf. Proposition 1.7, (vii)], we conclude that Π ∗ c diag ⊆ D c ∗ diag ∩ Π ∗ 2/1 ⊆ C Π ∗ 2 (Π ∗ c diag ) ∩ Π ∗ 2/1 = C Π ∗ 2/1 (Π ∗ c diag ) = Π ∗ c diag . This com- pletes the proof of assertion (i). Assertions (ii), (iii) follow immediately from assertion (i), together with the [easily verified] fact that the com- ) ∗ (p Π 2/1 posite D c ∗ diag → Π ∗ 2  Π ∗ 1 is surjective. Next, we verify assertion (iv). Since Π ∗ v new is slim and commensurably terminal in Π ∗ 2/1 [cf. Proposition 1.7, (ii), (vii)], it follows that I v ∗ new | z x ∩ Π ∗ 2/1 = {1}, which implies the injectivity of the composite in question. On the other hand, since the composite I v new | z x → D v new | z x → Π 2 | z x  Π z x is surjective [cf. [CbTpII], Lemma 3.11, (iv)], it follows immediately that the composite I v ∗ new | z x → D v ∗ new | z x → Π ∗ 2 | z x  Π ∗ z x is surjective. This completes the proof of assertion (iv). Next, we verify assertion (v). It follows immediately from asser- tion (iv), together with the commensurable terminality of Π ∗ v new in Π ∗ 2/1 [cf. Proposition 1.7, (vii)], that we have a natural exact sequence of profinite groups 1 −→ Π ∗ v new −→ D v ∗ new | z x −→ Π ∗ z x −→ 1 COMBINATORIAL ANABELIAN TOPICS III 49 — where we observe that the inclusion I v ∗ new | z x → D v ∗ new | z x determines a splitting of this exact sequence. Thus, it follows from the definition of I v ∗ new | z x that the natural inclusions Π ∗ v new , I v ∗ new | z x → D v ∗ new | z x deter- ∼ mine an isomorphism Π ∗ v new × I v ∗ new | z x → D v ∗ new | z x . On the other hand, again by the commensurable terminality of Π ∗ v new in Π ∗ 2/1 [cf. Proposi- tion 1.7, (vii)], the above displayed sequence implies that D v ∗ new | z x = C Π ∗ 2 | zx (Π ∗ v new ). This completes the proof of assertion (v). Next, we verify assertion (vi). It follows from the commensurable terminality of Π ∗ v new in Π ∗ 2/1 [cf. Proposition 1.7, (vii)] that D v ∗ new | z x ∩ Π ∗ 2/1 = Π ∗ v new . Thus, since Π ∗ 2/1 is normal in Π ∗ 2 , assertion (vi) follows immediately from [CbTpII], Lemma 3.9, (i). This completes the proof of assertion (vi). Finally, we verify assertion (vii). Since Π ∗ z x ⊆ Π ∗ 1 is commensu- rably terminal in Π ∗ 1 [cf. Proposition 1.7, (vii)], it follows from the surjectivity of the composite D v ∗ new | z x → Π ∗ 2 | z x  Π ∗ z x [cf. asser- tion (iv)] that C Π ∗ 2 (D v ∗ new | z x ) ⊆ Π ∗ 2 | z x . In particular, it follows im- mediately from assertions (v), (vi) that D v ∗ new | z x ⊆ C Π ∗ 2 (D v ∗ new | z x ) ⊆ C Π ∗ 2 (Π ∗ v new ) ∩ Π ∗ 2 | z x = C Π ∗ 2 | zx (Π ∗ v new ) = D v ∗ new | z x . This completes the proof of assertion (vii).  Lemma 2.12 (Commensurator of a tripod arising from an edge). In the notation of Lemma 2.11, let Π 2  Π ∗∗ 2 be an SA- maximal almost pro-l quotient of Π 2 [cf. Definition 2.1, (ii)] that dominates Π 2  Π ∗ 2 [cf. the discussion entitled “Topological groups” in §0]. We shall use similar notation ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ Π ∗∗ 2/1 ; Π 1\2 ; Π 1 ; Π {2} ; Π G ; Π G 2/1 ; ∗∗ ∗∗ ∗∗ Π ∗∗ ∗∗ ∗∗ (p Π 2/1 ) : Π 2  Π 1 ; (p 1\2 ) : Π 2  Π {2} ; ∗∗ ∗∗ ∗∗ ∗∗ Π ∗∗ z x ⊆ Π 1 ; Π c diag ⊆ Π v new ⊆ Π 2/1 ; ∗∗ ∗∗ ∗∗ Π ∗∗ 2 | z x ; D c diag ; I v new | z x ⊆ D v new | z x for objects associated to Π 2  Π ∗∗ 2 to the notation introduced in the statement of Lemma 2.11 for objects associated to Π 2  Π ∗ 2 . Suppose that the natural [outer] surjection Π 1  Π ∗∗ {2} dominates the quotient Π 1  Π ∗ 1 . Then the following hold: ∗ (i) The natural surjection Π ∗∗ 2  Π 2 determines a surjection ∗∗ ∗ I v new | z x  I v new | z x . ∗∗ (ii) The image of Z Π loc ∗∗ (Π ∗∗ v new ) ⊆ Π 2 [cf. the discussion entitled 2 “Topological groups” in [CbTpII], §0] in Π ∗ 2 coincides with I v ∗ new | z x . ∗∗ ∗ ∗ (iii) The image of C Π ∗∗ (Π ∗∗ v new ) ⊆ Π 2 in Π 2 is contained in D v new | z x . 2 50 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∗∗ (iv) The natural outer action, by conjugation, of N Π ∗∗ (Π ∗∗ v new ) ⊆ Π 2 2 ∗ on [not Π ∗∗ v new but] Π v new is trivial. Proof. First, we verify assertion (i). Observe that it is immediate that ∗ ∗ the image of I v ∗∗ new | z x ⊆ Π ∗∗ 2 in Π 2 is contained in I v new | z x . Thus, asser- tion (i) follows immediately from Lemma 2.11, (iv), together with the ∗ [easily verified] fact that the natural surjection Π ∗∗ 2  Π 2 determines a ∗ surjection Π ∗∗ z x  Π z x . This completes the proof of assertion (i). ∗ (Π ∗∗ Next, we verify assertion (ii). Write Im(Z Π loc ∗∗ v new )) ⊆ Π 2 for the 2 ∗∗ ∗ (Π ∗∗ image of Z Π loc ∗∗ v new ) ⊆ Π 2 in Π 2 . Then it follows from assertion (i) 2 (Π ∗∗ that I v ∗ new | z x ⊆ Im(Z Π loc ∗∗ v new )). Thus, to complete the verification of 2 ∗ (Π ∗∗ assertion (ii), it suffices to verify that Im(Z Π loc ∗∗ v new )) ⊆ I v new | z x . To 2 this end, let us observe that it follows immediately from the final por- ∗∗ ∗∗ tion of [CbTpII], Lemma 3.6, (iv), that the image (p Π 1\2 ) (Π v new ) ⊆ ∼ Π ∗∗ {2} coincides with the image of an edge-like subgroup of Π G ← Π 1 associated to z x ∈ Edge(G) via the natural [outer] surjection Π 1  Π ∗∗ {2} , hence that the image [which is well-defined up to conjugacy] of ∗∗ ∗∗ ∗∗ ∗ (p Π 1\2 ) (Π v new ) ⊆ Π {2} in Π 1 [where we recall that we have assumed ∗ that Π 1  Π ∗∗ {2} dominates Π 1  Π 1 ] is an edge-like subgroup of ∼ Π ∗G ← Π ∗ 1 associated to z x ∈ Edge(G). Thus, since every edge-like subgroup of Π ∗ 1 is commensurably terminal [cf. Proposition 1.7, (vii)], it follows that the image [which is well-defined up to conjugacy] of ∗∗ loc ∗∗ ∗ (p Π (Π ∗∗ v new )) ⊆ Π {2} in Π 1 is contained in an edge-like subgroup 1\2 ) (Z Π ∗∗ 2 ∼ of Π ∗G ← Π ∗ 1 associated to z x ∈ Edge(G). On the other hand, since loc loc Π ∗∗ ⊆ Π ∗∗ (Π ∗∗ (Π ∗∗ ) ⊆ C Π ∗∗ (Π ∗∗ ) = v new , we have Z Π ∗∗ v new ) ⊆ Z Π ∗∗ c diag c diag c diag 2 2 2 D c ∗∗ diag [cf. Lemma 2.11, (ii)]. In particular, it follows immediately ∗∗ loc ∗∗ from Lemma 2.11, (iii), that the image of (p Π (Π ∗∗ v new )) ⊆ Π 1 2/1 ) (Z Π ∗∗ 2 in Π ∗ 1 is contained in some Π ∗ 1 -conjugate of Π ∗ z x ⊆ Π ∗ 1 , hence [since ∗∗ (Π ∗∗ I v ∗∗ new | z x ⊆ Z Π loc ∗∗ v new ) surjects onto Π z x — cf. Lemma 2.11, (iv)] that 2 ∗∗ loc ∗∗ ∗ ∗ ∗ (Π ∗∗ the image of (p Π v new )) ⊆ Π 1 in Π 1 coincides with Π z x ⊆ Π 1 2/1 ) (Z Π ∗∗ 2 ∗ [cf. Proposition 1.7, (v)], i.e., Im(Z Π loc ∗∗ (Π ∗∗ v new )) ⊆ Π 2 | z x . Thus, since [as 2 loc ∗ (Π ∗∗ is easily verified] Im(Z Π loc ∗∗ v new )) ⊆ Z Π ∗ 2 (Π v new ), we conclude that 2 ∗ loc ∗ loc ∗ ∗ Im(Z Π loc ∗∗ (Π ∗∗ v new )) ⊆ Π 2 | z x ∩ Z Π ∗ 2 (Π v new ) = Z Π ∗ 2 | zx (Π v new ) = I v new | z x 2 [where the final equality follows from Lemma 2.11, (v), together with the slimness portion of Proposition 1.7, (ii)]. This completes the proof of assertion (ii). ∗ Next, we verify assertion (iii). Write Im(C Π ∗∗ (Π ∗∗ v new )) ⊆ Π 2 for the 2 ∗∗ ∗ (Π ∗∗ image of C Π ∗∗ v new ) ⊆ Π 2 in Π 2 . Then it follows from [CbTpII], 2 Lemma 3.9, (ii), that C Π ∗∗ (Π ∗∗ (Z Π loc ∗∗ (Π ∗∗ v new ) ⊆ N Π ∗∗ v new )); thus, it fol- 2 2 2 ∗∗ ∗∗ lows from assertion (ii) that Im(C Π 2 (Π v new )) ⊆ N Π 2 ∗ (I v ∗ new | z x ). In par- ticular, since D v ∗ new | z x is topologically generated by Π ∗ v new , I v ∗ new | z x [cf. COMBINATORIAL ANABELIAN TOPICS III 51 Lemma 2.11, (v)], we conclude that ∗ ∗ (Π ∗∗ Im(C Π ∗∗ v new )) ⊆ C Π ∗ 2 (D v new | z x ) = D v new | z x 2 [cf. Lemma 2.11, (vii)]. This completes the proof of assertion (iii). Assertion (iv) follows immediately from assertion (iii), together with Lemma 2.11, (v). This completes the proof of Lemma 2.12.  Corollary 2.13 (Almost pro-l quotients and tripod homomor- phisms). In the notation of Definition 2.1, suppose that n ≥ 3. Let Π tpd ⊆ Π 3 be a 1-central [{1, 2, 3}-]tripod of Π 3 [cf. [CbTpII], Defi- nitions 3.3, (i); 3.7, (ii)]; Π tpd  (Π tpd ) ‡ an almost pro-l quotient. Then the following hold: (i) There exists an F-characteristic SA-maximal almost pro- l quotient [cf. Definition 2.1, (ii), (iii)] Π ∗ n of Π n that satisfies the following condition: If we write Π ∗ 3 for the quotient of Π 3 determined by the quotient Π n  Π ∗ n and (Π tpd ) ∗ ⊆ Π ∗ 3 for the image of Π tpd ⊆ Π 3 in Π ∗ 3 , then the quotient Π tpd  (Π tpd ) ∗ dominates the quotient Π tpd  (Π tpd ) ‡ [cf. the discussion entitled “Topological groups” in §0]. (ii) Every element of the image Im(T Π tpd ) ⊆ Out(Π tpd ) of the tri- pod homomorphism T Π tpd : Out FC (Π n ) −→ Out C (Π tpd ) associated to Π n [cf. [CbTpII], Definition 3.19] preserves the kernel of the surjection Π tpd  (Π tpd ) ∗ of (i). Thus, we obtain a natural homomorphism Im(T Π tpd ) −→ Out((Π tpd ) ∗ ) . (iii) There exists an F-characteristic SA-maximal almost pro- ∗ l quotient Π n  Π ∗∗ n of Π n that dominates Π n  Π n [cf. (i)] such that the composite Out FC (Π n )  Im(T Π tpd ) → Out((Π tpd ) ∗ ) — where the first arrow is the homomorphism induced by T Π tpd ; the second arrow is the homomorphism of (ii) — factors through the natural surjection Out FC (Π n )  Out FC (Π ∗∗ n  Π n ) [cf. Definition 2.1, (viii); Remark 2.1.1]. Thus, we have a natural commutative diagram of profinite groups Out FC (Π n ) ⏐ ⏐  −−−→ Im(T Π tpd ) ⏐ ⏐  tpd ∗ ) ) . Out FC (Π ∗∗ n  Π n ) −−−→ Out((Π 52 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Proof. Assertion (i) is a consequence of Proposition 2.3, (iii). Asser- tion (ii) follows immediately from the fact that Π ∗ n is F-characteristic, together with the definition of T Π tpd . Finally, we verify assertion (iii). Let us first observe that it follows immediately from the definition of T Π tpd , together with Proposition 2.3, (ii) [where we observe that any closed subgroup of a finite product of almost pro-l groups is almost pro-l], that, to verify assertion (iii), it suffices to verify the following assertion: Claim 2.13.A: There exists an F-characteristic SA- maximal almost pro-l quotient Π 3  Π ∗∗ 3 of Π 3 that dominates Π 3  Π ∗ 3 such that if we write (Π tpd ) ∗∗ ⊆ tpd Π ∗∗ ⊆ Π 3 in Π ∗∗ 3 for the image of Π 3 , then any auto- tpd ∗ morphism of (Π ) determined by conjugating by an element γ ∗∗ ∈ N Π ∗∗ ((Π tpd ) ∗∗ ) 3 is (Π tpd ) ∗ -inner. To verify Claim 2.13.A, let Π 3  Π ∗∗ 3 be an F-characteristic SA- maximal almost pro-l quotient of Π 3 that dominates Π 3  Π ∗ 3 and ((Π tpd ) ∗∗ ). Then it follows immediately from [CmbCsp], γ ∗∗ ∈ N Π ∗∗ 3 Proposition 1.9, (i), that Z Π 3 (Π tpd ) ⊆ Π 3 surjects onto Π 1 , hence also ∗∗ onto Π ∗∗ 1 — where we write Π 1 for the quotient of Π 1 determined ∗∗ by the quotient Π 3  Π 3 . In particular, there exists an element τ ∈ Z Π 3 (Π tpd ) such that the images of γ ∗∗ and τ in Π ∗∗ 1 coincide. Thus, by replacing γ ∗∗ by the difference of γ ∗∗ and the image of τ in Π ∗∗ 3 , ∗∗ ∗∗ we may assume without loss of generality that γ ∈ Π 3/1 — where we write Π ∗∗ 3/1 for the quotient of Π 3/1 [cf. Definition 2.1] induced by the quotient Π 3  Π ∗∗ 3 . In particular, the existence of an F-characteristic SA-maximal almost pro-l quotient Π 3  Π ∗∗ 3 as in Claim 2.13.A follows immediately, in light of Proposition 2.3, (ii), from Lemma 2.12, (iv). This completes the proof of assertion (ii).  Finally, before proceeding, we review the following well-known result. Lemma 2.14 (Automorphisms of stable log curves). Let l be a def def prime number. Write l aut = l if l is odd; l aut = 4 if l is even. If G is a profinite group, then we shall refer to the tensor product with Z/l aut Z of the abelianization of G as the l aut -abelianization of G. Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0. (i) Let k be an algebraically closed field such that l is invertible in k, (Spec k) log the log scheme obtained by equipping Spec k with the log structure determined by the fs chart N → k that maps 1 → 0, X log a stable log curve over (Spec k) log , and α an automorphism of X log over (Spec k) log . Write Π 1 for COMBINATORIAL ANABELIAN TOPICS III 53 the maximal pro-l quotient of the kernel of the natural outer surjection π 1 (X log )  π 1 ((Spec k) log ). Suppose that α acts trivially on the l aut -abelianization of Π 1 . Then α is the identity automorphism. (ii) Write M log for the moduli stack of pointed stable curves of type (g, r) over Z[1/l], where we regard the marked points as unordered, equipped with the log structure determined by the divisor at infinity, and C log → M log for the tautological stable log curve over M log . Write N log → M log for the finite log étale morphism of log regular log stacks determined by the local system of trivializations of the l aut -abelianizations of the log fundamental groups of the various logarithmic fibers of C log → M log . Then the underlying algebraic stack N of N log is an algebraic space. Proof. First, we consider assertion (i). We begin by recalling that when X log is a smooth log curve, and r ≤ 1 [so g ≥ 1], assertion (i) follows immediately from classical theory of endomorphisms of semi-abelian va- rieties and automorphisms of stable curves [cf., e.g., [Des], Lemme 5.17; [DM], Theorems 1.11, 1.13], together with [in the case where l = 2] the well-known fact that every root of unity ζ such that (ζ − 1)/l aut is an algebraic integer is necessarily equal to 1. Now let us return to the case of an arbitrary stable log curve X log . Then it follows immediately from the description of the relationship between the abelianization of Π 1 and the abelianizations of verticial subgroups of Π 1 given in [NodNon], Lemma 1.4, together with the portion of assertion (i) that has already been verified, that α stabilizes and induces the identity automorphism on each of the irreducible components of X log of genus ≥ 1. Next, let us observe that it follows immediately from the definition of l aut , together with the well-known structure of the submodule of the abelianization of Π 1 generated by the cuspidal inertia subgroups, that α acts trivially on the set of cusps of X log . Thus, by considering the various connected components of the union of the genus zero irreducible components of X log , we conclude that, to complete the verification of assertion (i), it suffices to verify, in the case where g = 0, that any automorphism of X log over (Spec k) log that acts trivially on the set of cusps of X log is equal to the identity automorphism. But this follows immediately by induction on r, i.e., by considering, when r ≥ 4, the stable log curve obtained from X log by “forgetting”, successively, each of the cusps of X log . [Here, we apply the elementary combinatorial fact that every non-smooth pointed stable curve of genus 0 has at least two irreducible components that contain cusps.] This completes the proof of asser- tion (i). Assertion (ii) follows immediately from assertion (i), together with well-known generalities concerning algebraic stacks [cf., e.g., the discussion surrounding [FC], Chapter I, Theorem 4.10].  54 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI 3. Applications to the theory of tempered fundamental groups In the present §3, we apply the technical tools developed in the pre- ceding §2, together with the theory of [CbTpI], §5, to obtain applica- tions to the theory of tempered fundamental groups. In particular, we prove a generalization of a result due to André [cf. [André], Theorems 7.2.1, 7.2.3] concerning the characterization of the local Galois groups in the image of the outer Galois action associated to a hyperbolic curve over a number field [cf. Corollary 3.20, (iii), below]. Definition 3.1. Let n be a nonnegative integer. For  ∈ {◦, •}, let p be a prime number;  Σ a nonempty set of prime numbers such that  Σ  = {  p};  R a mixed characteristic complete discrete valuation ring of residue characteristic  p whose residue field is separably closed;  K the field of fractions of  R;  K an algebraic closure of  K. Write def I  K = Gal(  K/  K) for the absolute Galois group of  K;  R for the ∧ ∧ ring of integers of  K;  R for the  p-adic completion of  R;  K for ∧ the field of fractions of  R . If n ≥ 2, then we suppose further that  Σ is either equal to Primes or of cardinality one. Let  X  log K def be a smooth log curve over  K. Write X  log K = X  log K ×  K  K; (X  K ) log n for the n-th log configuration space [cf. the discussion entitled “Curves” in [CbTpII], §0] of the smooth log curve X  log K over  K. (i) We shall write   def Σ Π n = π 1 ((X  K ) log n ) for the maximal pro-  Σ quotient of the log fundamental group of (X  K ) log n . Thus, we have a natural outer Galois action  ρ n : I  K −→ Out(  Π n ) . Note that  Π n is equipped with a natural structure of pro-  Σ configuration space group [cf. [MzTa], Definition 2.3, (i)]. (ii) We shall write ∧  π 1 temp ((X  K ) log n ×  K K ) ∧  for the tempered fundamental group of (X  K ) log n ×  K K [cf. [André], §4]. [Here, we note that [André], §4, only discusses the ∧ case where the base field  K is a complete subfield of “C p ”. On the other hand, let us recall from [AbsTpI], Proposition 2.2, that any profinite group of GFG-type [cf. [AbsTpI], Definition COMBINATORIAL ANABELIAN TOPICS III 55 2.1, (i)] is topologically finitely generated, which implies that the set of open subgroups of a profinite group of GFG-type [such as  Π n ] is countable. In particular, one verifies easily [cf. also [Brk], Corollary 9.5, and the following discussion] that the construction of the tempered fundamental group given in ∧ [André], §4, applies even in the case where the base field  K is not a complete subfield of “C p ”.] We shall write  ∧  Π tp = lim π 1 temp ((X  K ) log n n ×  K K )/N ←− def N ∧  for the  Σ-tempered fundamental group of (X  K ) log n ×  K K [cf. [CmbGC], Corollary 2.10, (iii)], i.e., the inverse limit given by allowing N to vary over the open normal subgroups of  ∧ π 1 temp ((X  K ) log n ×  K K ) such that the quotient by N cor- responds to a topological covering [cf. [André], §4.2] of some  ∧ finite log étale Galois covering of (X  K ) log n ×  K K of degree a product of primes ∈  Σ. [Here, we recall that, when n = 1, such a “topological covering” corresponds to a “combinatorial covering”, i.e., a covering determined by a covering of the dual semi-graph of the special fiber of the stable model of some fi-  ∧ nite log étale covering of (X  K ) log n ×  K K .] Thus, we have a natural outer Galois action  tp ρ n : I  K −→ Out(  Π tp n ) [cf. [André], Proposition 5.1.1]. Lemma 3.2 (Pro-Σ completions of discrete free groups). Let Σ be a nonempty set of prime numbers and F a discrete free group. Then the following hold: (i) The natural homomorphism F → F Σ from F to the pro-Σ completion F Σ of F is injective. (ii) Suppose that F is not of rank one. Then the image of the injection F → F Σ of (i) is normally terminal [cf. the dis- cussion entitled “Topological groups” in [CbTpI], §0]. Proof. Assertion (i) follows immediately from [RZ], Proposition 3.3.15. Assertion (ii) follows immediately from the fact that F is conjugacy l-separable for every prime number l [cf. [Prs], Theorem 3.2], together with a similar argument to the argument applied in the proof of [André], Lemma 3.2.1. This completes the proof of Lemma 3.2.  Proposition 3.3 (Log and tempered fundamental groups). In the notation of Definition 3.1, the following hold: 56 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI  Σ (i) Write (  Π tp for the pro-  Σ completion of  Π tp n ) n . Then there  tp  Σ ∼  exists a natural outer isomorphism ( Π n ) → Π n .  (ii) The outer homomorphism  Π tp 1 → Π 1 determined by the outer isomorphism of (i) is injective.  (iii) The image of the outer injection  Π tp 1 → Π 1 of (ii) is nor- mally terminal. • tp ◦ • (iv) Write Isom( ◦ Π tp 1 , Π 1 ) (respectively, Isom( Π 1 , Π 1 )) for the ◦ • tp set of isomorphisms of ◦ Π tp 1 (respectively, Π 1 ) with Π 1 (re- spectively, • Π 1 ) and Inn(−) for the group of inner automor- phisms of “(−)”. Then the natural map between sets of outer isomorphisms [i.e., sets of “Inn(−)-orbits”] • tp • tp ◦ • • Isom( ◦ Π tp 1 , Π 1 )/Inn( Π 1 ) −→ Isom( Π 1 , Π 1 )/Inn( Π 1 ) induced by the natural outer isomorphism of (i) — hence also the natural homomorphism  Out(  Π tp 1 ) −→ Out( Π 1 ) — is injective. Proof. Assertion (i) follows immediately from the various definitions involved. Next, we verify assertion (ii) (respectively, (iii)). Let us first observe that it follows immediately from assertion (i) that, to verify assertion (ii) (respectively, (iii)), by replacing X  log K by a suitable con- nected finite log étale covering of X  log K , we may assume without loss of generality that the first Betti number of the dual semi-graph of the spe- cial fiber of the stable model of every connected finite log étale covering of X  log K is  = 1. Then since  Π tp 1 is a projective limit of extensions of finite groups whose orders are products of primes ∈  Σ by discrete free groups whose ranks are  = 1, assertion (ii) (respectively, (iii)) follows immediately from Lemma 3.2, (i) (respectively, (ii)). This completes the proof of assertion (ii) (respectively, (iii)). Assertion (iv) follows immediately from assertion (iii). This completes the proof of Proposi- tion 3.3.  Remark 3.3.1. The injections of Proposition 3.3, (iv), allow one to • tp • tp  tp regard Isom( ◦ Π tp 1 , Π 1 )/Inn( Π 1 ), (respectively, Out( Π 1 )) as a sub- set (respectively, subgroup) of Isom( ◦ Π 1 , • Π 1 )/Inn( • Π 1 ) (respectively, Out(  Π 1 )). Remark 3.3.2. The normal terminality of Proposition 3.3, (iii), may also be verified by applying the theory of [SemiAn] and [NodNon]. We refer to the proof of [IUTeichI], Proposition 2.4, (iii), for more details concerning this approach. COMBINATORIAL ANABELIAN TOPICS III 57 Definition 3.4. Let G be a [semi-]graph. Write Node(G) for the set of closed edges of G. Then we shall refer to a map def μ : Node(G) → R >0 = { a ∈ R | a > 0 } as a metric structure on G. Also, we shall refer to a [semi-]graph equipped with a metric structure as a metric [semi-]graph. Let Σ be a [possibly empty] set of prime numbers. Then we shall say that an ∼ isomorphism G 1 → G 2 between two [semi-]graphs G 1 , G 2 equipped with metric structures μ 1 , μ 2 is Σ-rationally compatible with the given metric structures if there exists an element def ξ ∈ ( Z Σ ) + (⊆ Q >0 = Q ∩ R >0 ) — i.e., a positive rational number that is invertible, as an integer, at the primes of Σ [cf. the notation of [CbTpI], Corollary 5.9, (iv), if def Σ  = ∅; set ( Z Σ ) + = Q >0 if Σ = ∅] — such that ξ · μ 1 coincides with the ∼ composite of the bijection Node(G 1 ) → Node(G 2 ) determined by the given isomorphism with μ 2 . [Thus, if G 1 = G 2 is a finite [semi-]graph, and μ 1 = μ 2 , then such a ξ is necessarily equal to 1. Alternatively, if Σ = Primes, then such a ξ is necessarily equal to 1.] Definition 3.5. In the notation of Definition 3.1, let Σ ⊆  Σ \ {  p} be a nonempty subset of  Σ \ {  p} and  H ⊆  Π 1 an open subgroup of  Π 1 . (i) We shall write G  H [Σ] for the semi-graph of anabelioids of pro-Σ PSC-type deter- mined by the special fiber [cf. [CmbGC], Example 2.5] of the stable model over  R of the connected finite log étale covering of X  log K corresponding to  H ⊆  Π 1 . (ii) We shall write G  H for the semi-graph associated to [i.e., the dual semi-graph of] the special fiber of the stable model over  R of the connected finite log étale covering of X  log K corresponding to  H ⊆  Π 1 , i.e., the underlying semi-graph of G  H [Σ] [cf. (i)]. Note that this semi-graph is independent of the choice of Σ. (iii) We shall write μ  H : Node(G  H ) −→ R >0 for the metric structure [cf. Definition 3.4] on G  H associated to the stable model over  R of the connected finite log étale 58 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI covering of X  log K corresponding to  H ⊆  Π 1 , i.e., the metric structure defined as follows: ∧ Write v  K ∧ for the  p-adic valuation of  K such that v  K ∧ (  p) = 1. Let e ∈ Node(G  H ). Suppose ∧ that the  R -algebra given by the completion at the node corresponding to e of the stable model of the connected covering of X  log K determined by  H ⊆  Π 1 is isomorphic to  ∧ R [[s 1 , s 2 ]]/(s 1 s 2 − a e ) ∧ — where a e ∈  R is a nonzero non-unit, and s 1 and def s 2 denote indeterminates. Then we set μ  H (e) = v  K ∧ (a e ): μ  H : Node(G  H ) −→ R >0 e → v  K ∧ (a e ) . Here, one verifies easily that “μ  H (a e )” depends only on e, i.e., is independent of the choice of the local equation “s 1 s 2 − a e ”. Remark 3.5.1. In the notation of Definition 3.5, it follows immedi- ately from the various definitions involved that one has a natural outer isomorphism ∼ (  H) Σ −→ Π G  H [Σ] between the maximal pro-Σ quotient (  H) Σ of  H and the [pro-Σ] fundamental group Π G  H [Σ] of the semi-graph of anabelioids of pro-Σ PSC-type G  H [Σ]. Proposition 3.6 (Equivalences of properties of isomorphisms between fundamental groups). In the notation of Definition 3.1, ∼ let α : ◦ Π 1 → • Π 1 be an isomorphism of profinite groups. [Thus, it follows immediately that ◦ Σ = • Σ — cf., e.g., the proof of [CbTpI], Proposition 1.5, (i).] Consider the following conditions: ∼ (a) The outer isomorphism ◦ Π 1 → • Π 1 determined by α is con- tained in • tp • tp ◦ • • Isom( ◦ Π tp 1 , Π 1 )/Inn( Π 1 ) ⊆ Isom( Π 1 , Π 1 )/Inn( Π 1 ) [cf. Remark 3.3.1], and ◦ Σ = • Σ ⊆ { ◦ p, • p}. (b ∀ ) For any characteristic open subgroup ◦ H ⊆ ◦ Π 1 of ◦ Π 1 and any nonempty subset Σ ⊆ ◦ Σ = • Σ such that ◦ p, • p ∈ Σ, if def we write • H = α( ◦ H) ⊆ • Π 1 , then the outer isomorphism ∼ ∼ of ( ◦ H) Σ → Π G ◦ H [Σ] [cf. Remark 3.5.1] with ( • H) Σ → Π G • H [Σ] COMBINATORIAL ANABELIAN TOPICS III 59 induced by α is group-theoretically verticial [cf. [CmbGC], Definition 1.4, (iv)]. (b ∃ ) For any characteristic open subgroup ◦ H ⊆ ◦ Π 1 of ◦ Π 1 , there exists a nonempty subset Σ ⊆ ◦ Σ = • Σ [which may depend on def ◦ H] such that ◦ p, • p ∈ Σ, and, moreover, if we write • H = ∼ α( ◦ H) ⊆ • Π 1 , then the outer isomorphism of ( ◦ H) Σ → Π G ◦ H [Σ] ∼ [cf. Remark 3.5.1] with ( • H) Σ → Π G • H [Σ] induced by α is group-theoretically verticial. (c ∀ ) For any characteristic open subgroup ◦ H ⊆ ◦ Π 1 of ◦ Π 1 and any nonempty subset Σ ⊆ ◦ Σ = • Σ such that ◦ p, • p ∈ Σ, if def we write • H = α( ◦ H) ⊆ • Π 1 , then the outer isomorphism of ∼ ∼ ( ◦ H) Σ → Π G ◦ H [Σ] [cf. Remark 3.5.1] with ( • H) Σ → Π G • H [Σ] induced by α is graphic [cf. [CmbGC], Definition 1.4, (i)]. (c ∃ ) For any characteristic open subgroup ◦ H ⊆ ◦ Π 1 of ◦ Π 1 , there exists a nonempty subset Σ ⊆ ◦ Σ = • Σ [which may depend on def ◦ H] such that ◦ p, • p ∈ Σ, and, moreover, if we write • H = ∼ α( ◦ H) ⊆ • Π 1 , then the outer isomorphism of ( ◦ H) Σ → Π G ◦ H [Σ] ∼ [cf. Remark 3.5.1] with ( • H) Σ → Π G • H [Σ] induced by α is graphic. Then: (i) We have implications: (b ∀ ) ⇐= (c ∀ ) ⇐= (c ∃ ) =⇒ (a) ⇐⇒ (b ∃ ) =⇒ (b ∀ ) . (ii) Suppose that ◦ Σ = • Σ ⊆ { ◦ p, • p}. [This condition is satisfied if, for instance, ◦ p = • p.] Then we have equivalences: (b ∃ ) ⇐⇒ (b ∀ ) and (c ∃ ) ⇐⇒ (c ∀ ) . (iii) Suppose that either ◦ p ∈ ◦ Σ or • p ∈ • Σ. Then we have equiva- lences: (a) ⇐⇒ (b ∃ ) ⇐⇒ (c ∃ ) . Moreover, (a), (b ∃ ), and (c ∃ ) imply that ◦ p = • p. Proof. First, we claim that the following assertion holds: Claim 3.6.A: Suppose that (a) is satisfied, and that either ◦ p ∈ ◦ Σ or • p ∈ • Σ. Then ◦ p = • p ∈ ◦ Σ = • Σ. Moreover, (c ∃ ) is satisfied. To verify Claim 3.6.A, suppose that (a) is satisfied, and that ◦ p ∈ ◦ Σ. Then it follows immediately from [SemiAn], Corollary 3.11 [cf., espe- cially, the portion of the statement and proof of [SemiAn], Corollary 3.11, concerning, in the notation of loc. cit., the assertion “p α = p β ”]; 60 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [SemiAn], Remark 3.11.1 [cf. also [AbsTpII], Lemma 2.6, (i); the state- ment and proof of [AbsTpII], Corollary 2.11; [AbsTpII], Remark 2.11.1, (i)], that ◦ p = • p ∈ ◦ Σ = • Σ, and, moreover, that (c ∃ ) is satisfied. This completes the proof of Claim 3.6.A. Next, we verify assertion (i). Let us first observe that it follows from the fact that graphicity implies group-theoretic verticiality that the following implications hold: (c ∀ ) ⇒ (b ∀ ) and (c ∃ ) ⇒ (b ∃ ). Next, we verify the implication (b ∃ ) ⇒ (b ∀ ) (respectively, (c ∃ ) ⇒ (c ∀ )). Suppose that (b ∃ ) (respectively, (c ∃ )) is satisfied. Then it follows that ◦ Σ = • Σ ⊆ { ◦ p, • p}. Next, let us observe that, to complete the verification of (b ∀ ) (respectively, (c ∀ )), we may assume without loss of generality — by replacing the open subgroup ◦ H ⊆ ◦ Π 1 in (b ∀ ) (respectively, (c ∀ )) by ◦ Π 1 — that ◦ H = ◦ Π 1 and • H = • Π 1 . Moreover, one verifies easily that, to complete the verification of (b ∀ ) (respectively, (c ∀ )), we may assume without loss of generality — by replacing the subset Σ in (b ∀ ) (respectively, (c ∀ )) by ◦ Σ\( ◦ Σ∩{ ◦ p, • p}) = • Σ\( • Σ∩{ ◦ p, • p}) ( = ∅) — that Σ = ◦ Σ \ ( ◦ Σ ∩ { ◦ p, • p}) = • Σ \ ( • Σ ∩ { ◦ p, • p}) ( = ∅). Let ◦ U ⊆ ◦ Π 1 def be a characteristic open subgroup. Write • U = α( ◦ U ) ⊆ • Π 1 . Then it follows immediately from (b ∃ ) (respectively, (c ∃ )) that there exists a nonempty subset Σ ◦ U ⊆ Σ such that α induces a functorial bijection ∼ Vert(G ◦ U [Σ]) = Vert(G ◦ U [Σ ◦ U ]) −→ Vert(G • U [Σ ◦ U ]) = Vert(G • U [Σ]) (respectively, ∼ VCN(G ◦ U [Σ]) = VCN(G ◦ U [Σ ◦ U ]) −→ VCN(G • U [Σ ◦ U ]) = VCN(G • U [Σ])). In particular, by considering these functorial bijections between the sets “Vert” (respectively, “VCN”) associated to the connected finite étale coverings corresponding to the various characteristic open sub- def groups ◦ U ⊆ ◦ Π 1 , • U = α( ◦ U ) ⊆ • Π 1 , we conclude that the isomor- ∼ • Σ phism ◦ Π Σ 1 → Π 1 is group-theoretically verticial (respectively, group- theoretically verticial and group-theoretically edge-like, hence graphic [cf. [CmbGC], Proposition 1.5, (ii)]). This completes the proof of the implication (b ∃ ) ⇒ (b ∀ ) (respectively, (c ∃ ) ⇒ (c ∀ )). Next, we observe that since (a) implies that ◦ Σ = • Σ ⊆ { ◦ p, • p}, the implication (a) ⇒ (b ∃ ) follows from [SemiAn], Theorem 3.7, (iv), together with [the evident Σ-tempered analogue of] the discussion of [SemiAn], Example 2.10. Thus, to complete the verification of assertion (i), it suffices to verify the implication (b ∃ ) ⇒ (a). To this end, suppose that (b ∃ ) is satisfied. Let ◦ H ⊆ ◦ Π 1 be a characteristic open subgroup of ◦ Π 1 . Then it follows from (b ∃ ) that there exists a nonempty subset def Σ ⊆ ◦ Σ = • Σ such that ◦ p, • p ∈ Σ, and, moreover, if we write • H = ∼ α( ◦ H) ⊆ • Π 1 , then the outer isomorphism of ( ◦ H) Σ → Π G ◦ H [Σ] [cf. ∼ Remark 3.5.1] with ( • H) Σ → Π G • H [Σ] induced by α is group-theoretically COMBINATORIAL ANABELIAN TOPICS III 61 verticial. For each  ∈ {◦, •}, write G  =c H [Σ] for the graph of anabelioids obtained by omitting the cusps [i.e., open edges] of G  H [Σ]; tp =c Π tp G  [Σ] , Π G  [Σ] H H for the tempered fundamental groups of G  H [Σ], G  =c H [Σ], respectively [cf. the discussion preceding [SemiAn], Proposition 3.6]. Here, let us observe that it follows immediately from the various definitions involved that we have a natural commutative diagram ∼ =c Π tp G  [Σ] −→ H ∩ Π tp G  [Σ] H ∩ ∼ ∼ =c Σ Σ (Π tp −→ (Π tp −→ Π G  H [Σ] G  [Σ] ) G  [Σ] ) H H =c tp Σ Σ — where we write (Π tp G  [Σ] ) , (Π G  [Σ] ) for the pro-Σ completions of H H =c tp Π tp G  H [Σ] , Π G  H [Σ] , respectively; the horizontal arrows are outer isomor- phisms; the lower right-hand horizontal arrow is the outer isomorphism of Proposition 3.3, (i); the vertical inclusions are the inclusions that arise from Proposition 3.3, (ii). ∼ ∼ Now since the outer isomorphism of ( ◦ H) Σ → Π G ◦ H [Σ] with ( • H) Σ → Π G • H [Σ] induced by α is group-theoretically verticial, it follows imme- diately from [NodNon], Proposition 1.13; the argument applied in the proof of the sufficiency portion of [CmbGC], Proposition 1.5, (ii), that ∼ =c α determines an isomorphism G ◦ =c H [Σ] → G • H [Σ] of graphs of anabe- lioids. Thus, it follows immediately from the existence of the natural outer isomorphisms discussed above that the [group-theoretically verti- ∼ cial] outer isomorphism Π G ◦ H [Σ] → Π G • H [Σ] induced by the isomorphism α maps the Π G ◦ H [Σ] -conjugacy class of Π tp G ◦ H [Σ] to the Π G • H [Σ] -conjugacy tp class of Π G • H [Σ] . Moreover, it follows immediately from the normal terminality of Proposition 3.3, (iii), that the resulting conjugacy inde- terminacies may be reduced to Π tp G  H [Σ] -conjugacy indeterminacies. In particular, by applying these observations to the various characteristic open subgroups “ ◦ H” of ◦ Π 1 , one verifies easily from the description of the tempered fundamental group as a [countably indexed!] projective limit given in [André], §4.5 [cf. also the discussion preceding [SemiAn], Proposition 3.6, as well as the discussion of Definition 3.1, (ii), of the ∼ present paper], that the outer isomorphism ◦ Π 1 → ◦ Π 1 determined by α • tp • tp ◦ • • is contained in Isom( ◦ Π tp 1 , Π 1 )/Inn( Π 1 ) ⊆ Isom( Π 1 , Π 1 )/Inn( Π 1 ), i.e., that (a) is satisfied. This completes the proof of the implication 62 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (b ∃ ) ⇒ (a), hence also of assertion (i). Assertion (ii) follows immedi- ately from assertion (i), together with the various definitions involved. Assertion (iii) follows from assertion (i), together with Claim 3.6.A. This completes the proof of Proposition 3.6.  Definition 3.7. In the notation of Definition 3.1: ∼ (i) Let α : ◦ Π 1 → • Π 1 be an isomorphism of profinite groups. Then we shall say that α is G-admissible [i.e., “graph-admissible”] if α satisfies condition (c ∃ ) — hence also conditions (a), (b ∀ ), (b ∃ ), (c ∀ ) [cf. Proposition 3.6, (i)] — of Proposition 3.6. Write Aut( ◦ Π 1 ) G ⊆ Aut( ◦ Π 1 ) for the subgroup [cf. the equivalence (c ∀ ) ⇔ (c ∃ ) of Proposi- tion 3.6, (ii))] of G-admissible automorphisms of ◦ Π 1 and Out( ◦ Π 1 ) G = Aut( ◦ Π 1 ) G /Inn( ◦ Π 1 ) ⊆ Out( ◦ Π 1 ) def for the subgroup of G-admissible outomorphisms of ◦ Π 1 . ∼ (ii) Let α : ◦ Π 1 → • Π 1 be an isomorphism of profinite groups [so ◦ Σ = • Σ — cf., e.g., the proof of [CbTpI], Proposition 1.5, (i)]. Let Σ ⊆ ◦ Σ = • Σ be a [possibly empty] subset such that ◦ p, • p ∈ Σ. Then we shall say that α is Σ-M-admissible [i.e., “Σ-metric-admissible”] if α is G-admissible [cf. (i)], and, moreover, the following condition is satisfied: Let ◦ H ⊆ ◦ Π 1 be a characteristic open subgroup of def ◦ Π 1 . Write • H = α( ◦ H) ⊆ • Π 1 . Then the isomor- phism of G ◦ H with G • H induced by α [where we note that one verifies easily that the isomorphism of G ◦ H with G • H induced by α does not depend on the choice of “Σ” in condition (c ∀ ) of Proposition 3.6] is Σ- rationally compatible [cf. Definition 3.4] with respect to the metric structures μ ◦ H , μ • H [cf. Definition 3.5, (iii)]. [Thus, if the collections of data labeled by ◦, • are equal, then the notion of Σ-M-admissibility is independent of the choice of Σ — cf. the final portion of Definition 3.4.] We shall say that α is M-admissible if α is ∅-M-admissible. Write Aut( ◦ Π 1 ) M ⊆ Aut( ◦ Π 1 ) for the subgroup of M-admissible automorphisms of ◦ Π 1 and Out( ◦ Π 1 ) M = Aut( ◦ Π 1 ) M /Inn( ◦ Π 1 ) ⊆ Out( ◦ Π 1 ) def for the subgroup of M-admissible outomorphisms of ◦ Π 1 . COMBINATORIAL ANABELIAN TOPICS III 63 (iii) We shall write Out F ( ◦ Π n ) M ⊆ Out F ( ◦ Π n ) for the subgroup of the group Out F ( ◦ Π n ) of F-admissible outo- morphisms of ◦ Π n [cf. [CmbCsp], Definition 1.1, (ii)] obtained by forming the inverse image of Out( ◦ Π 1 ) M ⊆ Out( ◦ Π 1 ) [cf. (ii)] via the natural homomorphism Out F ( ◦ Π n ) → Out F ( ◦ Π 1 ) = Out( ◦ Π 1 ) [cf. [CbTpI], Theorem A, (i)]; Out FC ( ◦ Π n ) M = Out F ( ◦ Π n ) M ∩ Out C ( ◦ Π n ) ⊆ Out FC ( ◦ Π n ) def [cf. [CmbCsp], Definition 1.1, (ii)]. Definition 3.8. In the notation of Definition 3.1: ∼ (i) Let α : ◦ Π n → • Π n be an isomorphism of profinite groups [so ◦ Σ = • Σ — cf., e.g., the proof of [CbTpI], Proposition 1.5, (i)] and l ∈ ◦ Σ = • Σ such that l ∈ { ◦ p, • p}. Then we shall say that α is {l}-I-admissible [i.e., “{l}-inertia-admissible”] if α is PF-admissible whenever n ≥ 2 [cf. [CbTpI], Definition 1.4, (i)], and, moreover, the following condition is satisfied: Let ◦ Π n  ( ◦ Π n ) ∗ be an F-characteristic almost pro-l quotient of ◦ Π n ( π 1 ((X ◦ K ) log n )) [cf. Definition 2.1, (iii)]. If ◦ Σ = • Σ  = Primes, then we assume fur- ther that the quotient ◦ Π n  ( ◦ Π n ) ∗ is an almost maximal pro-l quotient relative to some characteris- tic open subgroup of ◦ Π n [cf. Definition 1.1]. Write • Π n  ( • Π n ) ∗ for the quotient of • Π n that corre- sponds to ◦ Π n  ( ◦ Π n ) ∗ via α. [Here, we observe that since α is PF-admissible whenever n ≥ 2, one verifies immediately that the quotient • Π n  ( • Π n ) ∗ satisfies similar assumptions to the assumptions im- posed on the quotient ◦ Π n  ( ◦ Π n ) ∗ .] Then there exist open subgroups ◦ J ⊆ I ◦ K , • J ⊆ I • K [which may depend on ◦ Π n  ( ◦ Π n ) ∗ ] such that the diagram Im( ◦ J) −−−→ Out(( ◦ Π n ) ∗ ) ⏐ ⏐ ⏐ ⏐ β   Im( • J) −−−→ Out(( • Π n ) ∗ ) — where, for  ∈ {◦, •}, we write Im(  J) ⊆ Out((  Π n ) ∗ ) for the image of  J via the homomorphism  J → Out((  Π n ) ∗ ) induced [in light of our assumptions on 64 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI the quotients under consideration!] by  ρ n ; the hor- izontal arrows are the natural inclusions; the right- hand vertical arrow is the isomorphism induced by the isomorphism α — commutes for some [uniquely ∼ determined] isomorphism β : Im( ◦ J) → Im( • J). ∼ We shall say that an outer isomorphism ◦ Π n → • Π n is {l}-I- ∼ admissible if it arises from an isomorphism ◦ Π n → • Π n which is {l}-I-admissible. ∼ (ii) We shall say that an isomorphism of profinite groups ◦ Π n → • Π n [so ◦ Σ = • Σ — cf., e.g., the proof of [CbTpI], Proposition 1.5, (i)] is I-admissible [i.e., “inertia-admissible”] if ◦ Σ = • Σ ⊆ { ◦ p, • p}, and, moreover, the isomorphism is {l}-I-admissible [cf. (i)] for every prime number l ∈ ◦ Σ = • Σ such that l ∈ ∼ { ◦ p, • p}. We shall say that an outer isomorphism ◦ Π n → • Π n ∼ is I-admissible if it arises from an isomorphism ◦ Π n → • Π n which is I-admissible. (iii) Let l ∈ ◦ Σ be such that l  = ◦ p. Then we shall write Aut {l}-I ( ◦ Π n ) ⊆ Aut( ◦ Π n ) for the subgroup of {l}-I-admissible automorphisms of ◦ Π n [cf. (i)]; Out {l}-I ( ◦ Π n ) = Aut {l}-I ( ◦ Π n )/Inn( ◦ Π n ) ⊆ Out( ◦ Π n ) def for the subgroup of {l}-I-admissible outomorphisms of ◦ Π n ; Out F {l}-I ( ◦ Π n ) = Out {l}-I ( ◦ Π n ) ∩ Out F ( ◦ Π n ) ⊆ Out F ( ◦ Π n ) def [cf. [CmbCsp], Definition 1.1, (ii)]; Out FC {l}-I ( ◦ Π n ) = Out {l}-I ( ◦ Π n ) ∩ Out FC ( ◦ Π n ) ⊆ Out FC ( ◦ Π n ) def [cf. [CmbCsp], Definition 1.1, (ii)]. Also, we shall write  def Aut {l}-I ( ◦ Π n ) ⊆ Aut( ◦ Π n ) Aut I ( ◦ Π n ) = l∈ ◦ Σ\( ◦ Σ∩{ ◦ p}) for the subgroup of I-admissible automorphisms of ◦ Π n [cf. (ii)];  def Out I ( ◦ Π n ) = Out {l}-I ( ◦ Π n ) ⊆ Out( ◦ Π n ) l∈ ◦ Σ\( ◦ Σ∩{ ◦ p}) for the subgroup of I-admissible outomorphisms of ◦ Π n ; Out FI ( ◦ Π n ) = Out I ( ◦ Π n ) ∩ Out F ( ◦ Π n ) ⊆ Out F ( ◦ Π n ) ; def Out FCI ( ◦ Π n ) = Out I ( ◦ Π n ) ∩ Out FC ( ◦ Π n ) ⊆ Out FC ( ◦ Π n ) . def COMBINATORIAL ANABELIAN TOPICS III 65 (iv) Let l ∈ ◦ Σ be such that l  = ◦ p. Then we shall write Out F ( ◦ Π n ) {l}-I ⊆ Out F ( ◦ Π n ) for the subgroup of the group Out F ( ◦ Π n ) of F-admissible out- omorphisms of ◦ Π n obtained by forming the inverse image of Out {l}-I ( ◦ Π 1 ) ⊆ Out( ◦ Π 1 ) [cf. (iii)] via the natural homomor- phism Out F ( ◦ Π n ) → Out F ( ◦ Π 1 ) = Out( ◦ Π 1 ) [cf. [CbTpI], The- orem A, (i)]; Out FC ( ◦ Π n ) {l}-I = Out F ( ◦ Π n ) {l}-I ∩ Out C ( ◦ Π n ) ⊆ Out FC ( ◦ Π n ) . def Also, we shall write Out F ( ◦ Π n ) I = def  Out F ( ◦ Π n ) {l}-I ⊆ Out F ( ◦ Π n ) ; l∈ ◦ Σ\( ◦ Σ∩{ ◦ p}) Out FC ( ◦ Π n ) I = Out F ( ◦ Π n ) I ∩ Out C ( ◦ Π n ) ⊆ Out FC ( ◦ Π n ) . def Theorem 3.9 (Equivalence of metric-admissibility and inerti- a-admissibility). For  ∈ {◦, •}, let  p be a prime number;  Σ a nonempty set of prime numbers such that  Σ  = {  p};  R a mixed characteristic complete discrete valuation ring of residue characteristic  p whose residue field is separably closed;  K the field of fractions of  R;  K an algebraic closure of  K; X  log K a smooth log curve over  K. For  ∈ {◦, •}, write X  log K = X  log K ×  K  K ; def  def  Π 1 = π 1 (X  log K ) Σ for the maximal pro-  Σ quotient of the log fundamental group of X  log K . Let ∼ α : ◦ Π 1 → • Π 1 be an isomorphism of profinite groups. [Thus, it follows immediately that ◦ Σ = • Σ — cf., e.g., the proof of [CbTpI], Proposition 1.5, (i).] If ◦ p ∈ ◦ Σ and • p ∈ • Σ, then we assume further that α is group- theoretically cuspidal [cf. [CmbGC], Definition 1.4, (iv)]. Then the following conditions are equivalent: (a) α is M-admissible [cf. Definition 3.7, (ii)]. (b ∀ ) α is I-admissible [cf. Definition 3.8, (ii)]. (b ∃ ) There exists a prime number l ∈ ◦ Σ = • Σ such that l ∈ { ◦ p, • p}, and, moreover, α is {l}-I-admissible [cf. Defini- tion 3.8, (i)]. 66 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Proof. First, let us observe that it follows formally from the various definitions involved [cf. Definitions 3.7, (i), (ii); 3.8, (ii)] that conditions (a), (b ∀ ), and (b ∃ ) all imply that there exists a prime number l ∈ ◦ Σ = • Σ such that l ∈ { ◦ p, • p}. Now fix such a prime number l and consider the condition: (b {l} ): α is {l}-I-admissible [cf. Definition 3.8, (i)]. Then [since l is arbitrary, and condition (a) is manifestly independent of the choice of l] it follows formally from the various definitions involved that to verify Theorem 3.9, it suffices to verify the equivalence (a) ⇐⇒ (b {l} ) . To this end, let ◦ H ⊆ ◦ Π 1 be a characteristic open subgroup of ◦ Π 1 . def Write • H = α( ◦ H) ⊆ • Π 1 . Also, for each  ∈ {◦, •}, write (  Π 1 ) ∗ for the maximal almost pro-l quotient of  Π 1 with respect to  H. [Thus, ∼ (  H) {l} → Π G  H [{l}] ⊆ (  Π 1 ) ∗ — cf. Remark 3.5.1.] ∼ Next, let us observe that, for each  ∈ {◦, •}, since (  H) {l} → Π G  H [{l}] ⊆ (  Π 1 ) ∗ is open, and (  Π 1 ) ∗ is topologically finitely generated, slim [cf. Proposition 1.7, (i)] and almost pro-l, there exist an open subgroup  J ⊆ I  K of I  K and a homomorphism  ρ 1 [  H] :  J −→ Out((  H) {l} ) such that  ρ 1 [  H] is compatible [in the evident sense] with the homo- morphism  J → Out((  Π 1 ) ∗ ) induced by  ρ 1 : I  K → Out(  Π 1 ), and, moreover,  ρ 1 [  H] factors through the maximal pro-l quotient (  J) {l} of  J, which [as is easily verified] is isomorphic to Z l as an abstract profinite group. Moreover, it follows immediately from the various defi- nitions involved, together with the well-known properness of the moduli stack of pointed stable curves of a given type, that the outer represen- ∼ tation (  J) {l} → Out((  H) {l} ) → Out(Π G  H [{l}] ) arising from such a homomorphism  ρ 1 [  H] is of PIPSC-type [cf. Definition 1.3]. In par- ticular, it follows immediately from Theorem 1.11, (ii) [i.e., in essence, [CbTpII], Theorem 1.9, (ii)], that if α satisfies condition (b {l} ), i.e., α ∼ is {l}-I-admissible, then the isomorphism of ( ◦ H) {l} → Π G ◦ H [{l}] with ∼ ( • H) {l} → Π G • H [{l}] induced by α is group-theoretically verticial, hence also group-theoretically nodal. Thus, by allowing “  H” to vary among the various characteristic open subgroups of  Π 1 , we conclude that if α satisfies condition (b {l} ), i.e., α is {l}-I-admissible, then α satisfies condition (b ∃ ) of Proposi- tion 3.6, hence [cf. Proposition 3.6, (iii); our assumption that α is group-theoretically cuspidal if ◦ p ∈ ◦ Σ, • p ∈ • Σ] that α is G-admissible [cf. [CmbGC], Proposition 1.5, (ii)]. In particular, it follows from either ∼ of the conditions (a), (b {l} ) that the isomorphism of ( ◦ H) {l} → Π G ◦ H [{l}] ∼ with ( • H) {l} → Π G • H [{l}] induced by α is graphic [cf. the implication COMBINATORIAL ANABELIAN TOPICS III 67 (c ∃ ) ⇒ (c ∀ ) of Proposition 3.6, (i)], hence that α determines a commu- tative diagram of isomorphisms of profinite groups D G◦ [{l}]  Dehn(G ◦ H [{l}]) −−− H −−→ Node(G ◦ H [{l}]) Λ G ◦ H [{l}] ⏐ ⏐ ⏐ ⏐    −−−−−→ Dehn(G • H [{l}]) D G• H [{l}] Node(G • H [{l}]) Λ G • H [{l}] [cf. [CbTpI], Definition 4.4; [CbTpI], Theorem 4.8, (iv)]. On the other hand, since, for each  ∈ {◦, •}, the outer repre- ∼ sentation (  J) {l} → Out((  H) {l} ) → Out(Π G  H [{l}] ) is of PIPSC-type, it follows — by replacing  J by an open subgroup of  J if neces- sary — from [CbTpI], Corollary 5.9, (iii), that we may assume with- out loss of generality that this outer representation factors through Dehn(G  H [{l}]) ⊆ Out(Π G  H [{l}] ). Thus, by considering the Dehn co- ordinates [cf. [CbTpI], Definition 5.8, (i)] of the image of a topological generator of (  J) {l} in Dehn(G  H [{l}]) [with respect to a topological generator of Λ G  H [{l}] ], it follows immediately from [CbTpI], Theorem 5.7; [CbTpI], Lemma 5.4, (ii), together with the existence of the com- mutative diagram of the above display, that ∼ the isomorphism G ◦ H → G • H induced by α is ∅-ratio- nally compatible [cf. Definition 3.4] with the metric structures μ ◦ H , μ • H [cf. Definition 3.5, (iii)] if and only if the images of the homomorphisms ( ◦ J) {l} → Dehn(G ◦ H [{l}]) and ( • J) {l} → Dehn(G • H [{l}]) are com- patible, up to a Q >0 -multiple, with the isomorphisms induced by α. In particular, by applying this equivalence to the various characteris- tic open subgroups “  H”⊆  Π 1 of  Π 1 , we conclude that α satisfies condition (b {l} ), i.e., α is {l}-I-admissible, if and only if α satisfies condition (a), i.e., α is M-admissible. This completes the proof of The- orem 3.9.  Definition 3.10. In the notation of Definition 3.1, let l ∈  Σ be such that l  =  p and  H ⊆  Π n an open subgroup of  Π n . For each i ∈ {0, · · · , n}, write  H i ⊆  Π i for the open subgroup of the quotient log  Π n   Π i [induced by the projection (X  K ) log n → (X  K ) i to the first log i factors] determined by the image of  H ⊆  Π n ;  Y i → (X  K ) log i for the connected finite log étale covering of (X  K ) log corresponding to i  H i ⊆  Π i . Then we have a sequence of morphisms of log schemes  log log log log log Y n −→  Y n−1 −→ · · · −→  Y 2 −→  Y 1 −→  Y 0 . 68 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI log Thus, for i ∈ {0, · · · , n}, if we write  U i for the interior of  Y i [cf. the discussion entitled “Log schemes” in [CbTpI], §0], we obtain a sequence of morphisms of schemes [each of which determines a family of hyperbolic curves]  U n −→  U n−1 −→ · · · −→  U 2 −→  U 1 −→  U 0 . Then we shall say that  H is of l-polystable type if the following con- ditions are satisfied: (a) For each i ∈ {0, · · · , n}, α ∈ Aut F (  Π i ) [cf. [CmbCsp], Defi- nition 1.1, (ii)], the open subgroup  H i ⊆  Π i is preserved by α. Here, for convenience, when n = 1, and  Σ is arbitrary, we def set Aut F (  Π 1 ) = Aut(  Π 1 ). [In particular,  H i is normal.] (b) The [necessarily F-characteristic — cf. condition (a) above; Definition 2.1, (iii)] maximal almost pro-l quotient   ∗ (π 1 ((X  K ) log n ) ) Π n  ( Π n ) with respect to  H =  H n ⊆  Π n [cf. Definition 1.1] is SA- maximal [cf. Definition 2.1, (ii)]. (c) For each i ∈ {1, · · · , n}, if we write (  H i/i−1 ) {l} for the maxi- def mal pro-l quotient of the kernel  H i/i−1 = Ker(  H i   H i−1 ), then the natural action of  H i−1 on the l aut -abelianization [cf. Lemma 2.14] of (  H i/i−1 ) {l} is trivial. Remark 3.10.1. In the notation of Definition 3.10: (i) Let us observe that [one verifies easily that] condition (c) of Definition 3.10 implies that the following condition holds: (d) For each i ∈ {1, · · · , n}, the natural outer representation  H i−1 −→ Out((  H i/i−1 ) {l} ) factors through a pro-l quotient of  H i−1 . Moreover, it follows from Lemma 2.14, (ii); [ExtFam], Corol- lary 7.4 [together with the well-known structure of the sub- module of the abelianization of (  H i/i−1 ) {l} generated by the cuspidal inertia subgroups — cf. the proof of Lemma 2.14, (i)], that condition (c) of Definition 3.10 also implies that the fol- lowing condition holds: (e) The sequence of morphisms of log schemes in Definition 3.10  log log log log log Y n −→  Y n−1 −→ · · · −→  Y 2 −→  Y 1 −→  Y 0 COMBINATORIAL ANABELIAN TOPICS III 69 extends to the factorization  log log log log log Y n −→  Y n−1 −→ · · · −→  Y 2 −→  Y 1 −→  Y 0 log associated to the base-change to  Y 0 of the log polystable morphism determined by a [uniquely determined!] stable polycurve over the integral closure of  R in some finite subextension of  K in  K [cf. [ExtFam], Definition 4.5]. log Here, the log structure of  Y 0 is the log structure on  Y 0 = Spec  R determined by the multiplicative monoid of nonzero elements of  R. (ii) One verifies easily that, for each i ∈ {0, · · · , n}, if  H ⊆  Π n is of l-polystable type, then  H i ⊆  Π i is of l-polystable type. Next, we define the notion of an  H-l-system. Roughly speaking, the notion of an  H-l-system may be understood as a basis for the topology of the maximal pro-l-quotient of an open subgroup  H ⊆  Π n consisting of open subgroups ⊆  Π n that, together with  H itself, correspond to coverings that are sufficiently well-behaved in various technical respects to allow us to apply to them the theory developed thus far in the present paper. Definition 3.11. In the notation of Definition 3.10, suppose that  H is of l-polystable type [cf. Definition 3.10]. (i) We shall write VCN sch (  H) for the set of points y ∈  Y n of the underlying scheme  Y n of  log Y n [cf. the notation of condition (e) of Remark 3.10.1, (i)] that satisfy the following condition: For i ∈ {0, · · · , n}, write def log y i ∈  Y i for the image of y in  Y i and y i log =  Y i ×  Y i y i . [Thus, for each i ∈ {1, · · · , n}, we have a stable log curve def log log log  log .] Then Y i | y log =  Y i ×  Y log y i−1 over y i−1 i−1 i−1 (a) y 0 is the closed point of  Y 0 = Spec  R; log (b) for each i ∈ {1, · · · , n}, the point of  Y i | y log determined i−1 by y i log is either a cusp, node, or generic point [i.e., the generic point of an irreducible component] of the stable log log curve  Y i | y log . i−1 Moreover, we shall write VCN sch (  H) for the set of elements y ∈ VCN sch (  H) such that, in the above notation, 70 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (c) for each i ∈ {1, · · · , n}, the residue field k(y i−1 ) of y i−1 is separably closed in the residue field k(y i ) of y i . [Here, we note that this separably closedness assumption means that the element ∈ VCN sch (  H) under consideration corre- sponds to a single element — i.e., as opposed to a Galois orbit of elements! — of the set of cusps/nodes/vertices in the semi- graph determined by the stable log curve fiber considered in (b).] Thus, for each i ∈ {1, · · · , n}, we have natural maps VCN sch (  H)  VCN sch (  H i ), VCN sch (  H) → VCN sch (  H i ), the first of which is surjective. Finally, we shall say that  H is VCN-complete if the equality VCN sch (  H) = VCN sch (  H) holds. (ii) We shall refer to a projective system  H = {  H λ } λ∈Λ of open subgroups of  Π n as an  H-l-system if each  H λ is of l-polystable type, VCN-complete, and contained in  H [i.e.,  H λ ⊆  H], and, moreover,         Ker  H  (  H) {l} = Ker  Π n  (  Π n ) ∗ = H λ λ∈Λ  [cf. condition (b) of Definition 3.10], i.e., the system H arises from a basis of the topology of (  H) {l} . (iii) Let  H = {  H λ } λ∈Λ be an  H-l-system [cf. (ii)]. Then we shall write VCN sch (  H) = lim VCN sch (  H λ ) ←− def λ∈Λ [cf. (i) above; the portion of [ExtFam], Corollary 7.4, concern- ing extensions of morphisms; our assumption that each  H λ arises from an open subgroup of (  H) {l} , where l  =  p]. In fact, we shall see below that VCN sch (  H) is independent of the choice of  H [cf. Lemma 3.14, (iv)]. Here, we note that one verifies easily that, for each i ∈ {0, · · · , n}, if  H = {  H λ } λ∈Λ is an  H-l-system, and we write (  H λ ) i ⊆  Π i for the image of def  H λ in  Π i , then the system  H i = {(  H λ ) i } λ∈Λ is an  H i - l-system [cf. (i), (ii) above; condition (b) of Definition 3.10; Remark 3.10.1, (ii)]. Thus, for each i ∈ {0, · · · , n}, we have a natural map VCN sch (  H) −→ VCN sch (  H i ) . Definition 3.12. In the notation of Definition 3.11, let  H = {  H λ } λ∈Λ be an  H-l-system [cf. Definition 3.11, (ii)] and y  ∈ VCN sch (  H) [cf. COMBINATORIAL ANABELIAN TOPICS III 71 Definition 3.11, (iii)]. For each i ∈ {0, · · · , n}, write y  i ∈ VCN sch (  H i ) for the image of y  via the natural map of the final display of Defini- tion 3.11, (iii). Let i ∈ {1, · · · , n}. (i) Write G i, y i−1 for the semi-graph of anabelioids of pro-l PSC-type determined by the stable log curve constituted by the log geometric fiber log log log of  Y i →  Y i−1 [cf. Definition 3.11, (i)] at the point of  Y i−1 determined by y  i−1 ; G  i, y −→ G i, y i−1 i−1 for the universal covering [corresponding to the [pro-l] funda- mental group Π G i, yi−1 of G i, y i−1 relative to the basepoint of G i, y i−1 determined by the various  H λ ’s] obtained by considering the “G i, y i−1 ’s” arising from the various  H λ ’s. (ii) Write  VCN sch (  H i )| y  i−1 = { y   ∈ VCN sch (  H i ) | y  i−1 = y  i−1 } def [cf. Definition 3.11, (iii)]. Then one verifies easily from the various definitions involved that we have a natural bijection ∼ VCN sch (  H i )| y  −→ VCN( G  i, y ) i−1 i−1 [cf. (i)]. In particular, the element y  i ∈ VCN sch (  H i )| y  i−1 determines an element z  i, y ∈ VCN( G  i, y ) i−1 of VCN( G  i, y i−1 ). (iii) It follows immediately from the various definitions involved that we have a natural action of (  H i ) {l} , hence also of (  H i/i−1 ) {l} [cf. the notation of condition (c) of Definition 3.10], on the set VCN sch (  H i ). Thus, we obtain a tautological isomorphism ∼ Π G i, yi−1 −→ (  H i/i−1 ) {l} such that the various VCN-subgroups [cf. [CbTpI], Definition 2.1, (i)] on the left-hand side of this isomorphism correspond to the various stabilizer subgroups of (  H i/i−1 ) {l} associated to elements of VCN sch (  H i )| y  i−1 [cf. the notation of (ii); the natural bijection of the second display of (ii)] on the right-hand side of this isomorphism. (iv) Let (F i ) i∈{1,··· ,n} be a collection of closed subgroups F i ⊆ (  H i ) {l} . Then we shall say that the collection (F i ) i∈{1,··· ,n} is the VCN- chain of  H associated to y  ∈ VCN sch (  H) if, for each i ∈ {1, · · · , n}, the closed subgroup F i coincides with the image of 72 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI the VCN-subgroup of Π G i, yi−1 associated to z  i, y ∈ VCN( G  i, y i−1 ) ∼ [cf. (ii)] via the isomorphism Π G i, yi−1 → (  H i/i−1 ) {l} ⊆ (  H i ) {l} of (iii). We shall say that the collection (F i ) i∈{1,··· ,n} is an  H-VCN-chain of  H if (F i ) i∈{1,··· ,n} is the VCN-chain of  H associated to an element of VCN sch (  H). Write VCN gp (  H) for the set of  H-VCN-chains of  H. In fact, we shall see be- low that VCN gp (  H) is independent of the choice of  H [cf. Lemma 3.14, (iv)]. Thus, we conclude from [CmbGC], Propo- sition 1.2, (i), that the natural bijections of (ii) determine a bijection ∼ VCN sch (  H) −→ VCN gp (  H) . Definition 3.13. In the notation of Definition 3.1: ∼ (i) We shall say that an isomorphism of profinite groups ◦ Π n → • Π n is SAF-admissible [i.e., “standard-adjacent-fiber-admissi- ble”] if it is PF-admissible whenever n ≥ 2 [cf. [CbTpI], Defi- nition 1.4, (i)] and, moreover, is compatible with the standard fiber filtrations on ◦ Π n and • Π n [cf. [CmbCsp], Definition 1.1, ∼ (i)]. We shall refer to an outer isomorphism ◦ Π n → • Π n as SAF-admissible if it arises from an SAF-admissible isomor- phism. One verifies easily that, in the case of an automor- phism or outomorphism, SAF-admissibility is equivalent to F- admissibility whenever n ≥ 2. ∼ (ii) Let α : ◦ Π n → • Π n be an isomorphism of profinite groups [so ◦ Σ = • Σ — cf., e.g., the proof of [CbTpI], Proposition 1.5, (i)] and l ∈ ◦ Σ = • Σ such that l ∈ { ◦ p, • p}. Then we shall say that α is {l}-G-admissible [i.e., {l}-graph-admissible] if α is SAF- admissible [cf. (i)], and, moreover, the following condition is satisfied: Let ◦ J ⊆ ◦ Π n be an open subgroup of ◦ Π n . Then there exist an open subgroup ◦ H ⊆ ◦ Π n of ◦ Π n of l-polystable type [cf. Definition 3.10] and an ◦ H-l- system ◦ H = { ◦ H λ } λ∈Λ [cf. Definition 3.11, (ii)] such def that ◦ H ⊆ ◦ J, • H = α( ◦ H) is of l-polystable type, def • H = { • H λ = α( ◦ H λ )} λ∈Λ is an • H-l-system, and, ∼ moreover, the isomorphism ◦ H → • H determined by α induces a bijection ∼ VCN gp ( ◦ H) −→ VCN gp ( • H) [cf. Definition 3.12, (iv)]. COMBINATORIAL ANABELIAN TOPICS III 73 ∼ We shall say that an outer isomorphism ◦ Π n → • Π n is {l}-G- admissible if it arises from an {l}-G-admissible isomorphism. ∼ (iii) We shall say that an isomorphism ◦ Π n → • Π n [so ◦ Σ = • Σ — cf., e.g., the proof of [CbTpI], Proposition 1.5, (i)] is G- admissible [i.e., graph-admissible] if ◦ Σ = • Σ ⊆ { ◦ p, • p}, and, moreover, the isomorphism is {l}-G-admissible [cf. (ii)] for every prime number l ∈ ◦ Σ = • Σ such that l ∈ { ◦ p, • p}. We ∼ shall say that an outer isomorphism ◦ Π n → • Π n is G-admissible if it arises from a G-admissible isomorphism. (iv) We shall write Aut {l}-G ( ◦ Π n ) ⊆ Aut( ◦ Π n ) for the subgroup [cf. Lemma 3.14, (ii), (iii), below] of {l}-G- admissible automorphisms of ◦ Π n [cf. (ii)]; Out {l}-G ( ◦ Π n ) = Aut {l}-G ( ◦ Π n )/Inn( ◦ Π n ) ⊆ Out( ◦ Π n ) def for the subgroup of {l}-G-admissible outomorphisms of ◦ Π n ;  def Aut G ( ◦ Π n ) = Aut {l}-G ( ◦ Π n ) ⊆ Aut( ◦ Π n ) l∈ ◦ Σ\( ◦ Σ∩{ ◦ p}) for the subgroup of G-admissible automorphisms of ◦ Π n [cf. (iii)];  def Out G ( ◦ Π n ) = Out {l}-G ( ◦ Π n ) ⊆ Out( ◦ Π n ) l∈ ◦ Σ\( ◦ Σ∩{ ◦ p}) for the subgroup of G-admissible outomorphisms of ◦ Π n . Remark 3.13.1. (i) In the notation of Definition 3.13, suppose that n = 1. Then it follows immediately from Proposition 3.6, (ii); Lemma 3.14, (ii), (iii), below; [CmbGC], Proposition 1.5, (ii), that the fol- lowing conditions are equivalent: • α is G-admissible in the sense of Definition 3.7, (i). • There exists a prime number l ∈ ◦ Σ = • Σ such that l ∈ { ◦ p, • p}, and, moreover, α is {l}-G-admissible in the sense of Definition 3.13, (ii). • α is G-admissible in the sense of Definition 3.13, (iii). In particular, for any prime number l ∈ ◦ Σ such that l  = ◦ p, we have equalities Out( ◦ Π 1 ) G = Out G ( ◦ Π 1 ) = Out {l}-G ( ◦ Π 1 ) [cf. Definitions 3.7, (i); 3.13, (iv)]. 74 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (ii) In the notation of Definition 3.13, (iv), one verifies easily from the various definitions involved that Out G ( ◦ Π n ) ⊆ Out {l}-G ( ◦ Π n ) ⊆ Out FC ( ◦ Π n ) [cf. [CmbCsp], Definition 1.1, (ii)]. Lemma 3.14 (Subgroups of l-polystable type). In the notation of ∼ Definition 3.1, let α : ◦ Π n → • Π n be an isomorphism of profinite groups [so ◦ Σ = • Σ — cf., e.g., the proof of [CbTpI], Proposition 1.5, (i)] and l ∈ ◦ Σ = • Σ such that l ∈ { ◦ p, • p}. Suppose that α is SAF-admissible [cf. Definition 3.13, (i)]. Then the following hold: (i) Let ◦ J ⊆ ◦ Π n be an open subgroup of ◦ Π n . Then there exists an open subgroup ◦ H ⊆ ◦ Π n of ◦ Π n of l-polystable type [cf. Definition 3.10] such that ◦ H ⊆ ◦ J. (ii) Let ◦ H ⊆ ◦ Π n be an open subgroup of ◦ Π n of l-polystable def type. Then • H = α( ◦ H) is an open subgroup of • Π n of l- polystable type. (iii) Let ◦ H ⊆ ◦ Π n be an open subgroup of l-polystable type of ◦ Π n . Then there exists an ◦ H-l-system ◦ H = { ◦ H λ } λ∈Λ [cf. def Definition 3.11, (ii)] such that • H = { • H λ = α( ◦ H λ )} λ∈Λ is an • H-l-system [cf. (ii)]. (iv) Let ◦ H ⊆ ◦ Π n be an open subgroup of l-polystable type of ◦ Π n ; ◦ H = { ◦ H λ } λ∈Λ , ◦ H † = { ◦ H λ † † } λ † ∈Λ † ◦ H-l-systems. Then there exists an ◦ H-l-system ◦ H ‡ = { ◦ H λ ‡ ‡ } λ ‡ ∈Λ ‡ that satisfies the condition that, for each (λ, λ † ) ∈ Λ × Λ † , there ex- ists a λ ‡ ∈ Λ ‡ such that ◦ H λ ‡ ‡ ⊆ ◦ H λ ∩ ◦ H λ † † . In particular, the sets VCN sch (  H) [cf. Definition 3.11, (iii)] and VCN gp (  H) [cf. Definition 3.12, (iv)] are independent of the choice of  H [cf. Definition 3.11, (ii)], i.e., depend only on ◦ H, re- spectively. (v) Let ◦ H, ◦ H † ⊆ ◦ Π n be open subgroups of l-polystable type of ◦ Π n ; ◦ H = { ◦ H λ } λ∈Λ an ◦ H-l-system; ◦ H † = { ◦ H λ † † } λ † ∈Λ † an ◦ H † -l-system. Suppose that the inclusion ◦ H † ⊆ ◦ H, hence def def also the inclusion • H † = α( ◦ H † ) ⊆ • H = α( ◦ H), holds. def Suppose, moreover, that • H = { • H λ = α( ◦ H λ )} λ∈Λ is an • H- def l-system [cf. (ii)], and that • H † = { • H λ † † = α( ◦ H † λ † )} λ † ∈Λ † is ∼ an • H † -l-system [cf. (ii)]. Then if the isomorphism ◦ H † → • † H determined by α induces a bijection ∼ VCN gp ( ◦ H † ) −→ VCN gp ( • H † ), COMBINATORIAL ANABELIAN TOPICS III 75 ∼ then the isomorphism ◦ H → • H determined by α induces a bijection ∼ VCN gp ( ◦ H) −→ VCN gp ( • H). Proof. First, we verify assertion (i) by induction on n. Write ◦ J n−1 for the image of ◦ J in ◦ Π n−1 and ( ◦ J n/n−1 ) {l} for the maximal pro-l def quotient of the kernel ◦ J n/n−1 = Ker( ◦ J  ◦ J n−1 ). Now let us observe that if n = 1, then assertion (i) follows immediately from the various definitions involved [cf. also the fact that ◦ Π n is topologically finitely generated — cf. [MzTa], Proposition 2.2, (ii)]]. Thus, suppose that n ≥ 2, and that the induction hypothesis is in force. Next, let us observe that since ◦ Π n is topologically finitely generated [cf. [MzTa], Proposition 2.2, (ii)], we may assume without loss of gen- erality — by replacing ◦ J by a suitable open subgroup of ◦ J — that ◦ J satisfies condition (a) of Definition 3.10 in the case where we take “i” to be n. Next, by applying the induction hypothesis to ◦ J n−1 , we obtain an open subgroup ◦ H n−1 ⊆ ◦ Π n−1 of ◦ Π n−1 that is contained in def ◦ J n−1 and of l-polystable type. Write ◦ H = ◦ H n−1 × ◦ J n−1 ◦ J. Thus, we have an exact sequence of profinite groups 1 −→ ◦ J n/n−1 −→ ◦ H −→ ◦ H n−1 −→ 1. Then it follows immediately from the condition imposed above on ◦ J, together with the induction hypothesis, that [by taking ◦ J n−1 to be sufficiently small] we may assume without loss of generality that ◦ H satisfies conditions (a) and (c) of Definition 3.10 [hence also (d) of Remark 3.10.1, (i)]. On the other hand, by considering the quotient ◦ out H  ( ◦ J n/n−1 ) {l}  ( ◦ H n−1 ) {l} [i.e., that arises from the fact that ◦ H satisfies condition (d) of Remark 3.10.1, (i) — cf. also the discussion entitled “Topological groups” in [CbTpI], §0], we conclude that the natural homomorphism ( ◦ J n/n−1 ) {l} → ( ◦ H) {l} induced by the natural inclusion ◦ J n/n−1 → ◦ H is injective. Thus, one verifies easily from Lemma 1.2, (i) [where we take “(G, N, J)” to be ( ◦ Π n , ◦ H, ◦ Π n−1 )], (ii) [where we take “(G, N, H)” to be ( ◦ Π n , ◦ H, ◦ Π n/n−1 )], together with our choice of ◦ H n−1 , that ◦ H satisfies condition (b) of Definition 3.10, i.e., that ◦ H is l-polystable type. This completes the proof of assertion (i). Assertion (ii) follows immediately from the various definitions involved. Next, we verify assertions (iii), (iv). Let us first observe that, to verify assertions (iii), (iv), it suffices to verify the following assertion: Claim 3.14.A: Let ◦ J ⊆ ◦ H be an open subgroup that arises from an open subgroup of the maximal pro-l quotient ( ◦ H) {l} of ◦ H. Then there exists an open def subgroup ◦ N ⊆ ◦ J such that ◦ N , • N = α( ◦ N ) are of l-polystable type, VCN-complete, and, moreover, arise from open subgroups of ( ◦ H) {l} , ( • H) {l} , respectively. 76 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI In the remainder of the proofs of assertions (iii), (iv), we verify Claim def 3.14.A by induction on n. Write • J = α( ◦ J). For each  ∈ {◦, •}, write  J n−1 for the image of  J in  Π n−1 and (  J n/n−1 ) {l} for the max- def imal pro-l quotient of the kernel  J n/n−1 = Ker(  J   J n−1 ). Now let us observe that if n = 1, then Claim 3.14.A follows immediately from the various definitions involved [cf. also the fact that ◦ Π n is topo- logically finitely generated — cf. [MzTa], Proposition 2.2, (ii)]]. Thus, suppose that n ≥ 2, and that the induction hypothesis is in force. Next, let us observe that since ◦ Π n is topologically finitely generated [cf. [MzTa], Proposition 2.2, (ii)], and ◦ H satisfies condition (a) of Def- inition 3.10, we may assume without loss of generality — by replacing ◦ J by a suitable open subgroup of ◦ J — that ◦ J satisfies condition (a) of Definition 3.10 in the case where we take “i” to be n. Next, let us observe that since ◦ J arises from an open subgroup of ( ◦ H) {l} , by con- ∼ out sidering the natural isomorphism ( ◦ H) {l} → ( ◦ H n/n−1 ) {l}  ( ◦ H n−1 ) {l} [i.e., that arises from the fact that ◦ H satisfies condition (d) of Re- mark 3.10.1, (i)], we conclude that ◦ J satisfies condition (d) of Re- mark 3.10.1, (i), in the case where we take “i” to be n. In particular, since the natural action of ◦ J n−1 on (( ◦ J n/n−1 ) {l} ) ab ⊗ Z Z/l aut Z factors through a pro-l quotient of ◦ J n−1 , we may assume without loss of gen- erality — by replacing ◦ J by the inverse image in ◦ J of a suitable open subgroup of ◦ J n−1 — that ◦ J satisfies condition (c) of Definition 3.10 in the case where we take “i” to be n. Thus, by applying the induction hypothesis to ◦ J n−1 ⊆ ◦ H n−1 , we def obtain an open subgroup ◦ N n−1 ⊆ ◦ J n−1 such that ◦ N n−1 , • N n−1 = α( ◦ N n−1 ) are of l-polystable type, VCN-complete, and arise from open subgroups of ( ◦ H n−1 ) {l} , ( • H n−1 ) {l} , respectively. Write ◦ N = ◦ N n−1 × ◦ J n−1 ◦ J. def Then one verifies immediately, by a similar argument to the argument def applied in the final portion of the proof of assertion (i), that ◦ N , • N = α( ◦ N ) are of l-polystable type and, moreover, arise from open subgroups of ( ◦ H) {l} , ( • H) {l} , respectively. In particular, since ◦ N , • N satisfy condition (d) of Remark 3.10.1, (i), in the case where we take “i” to be n, we may assume without loss of generality — by replacing ◦ N by the inverse image in ◦ N of a suitable open subgroup of ◦ N n−1 [cf. the induction hypothesis] — that ◦ N , • N satisfy the condition that each of the elements of VCN sch ( ◦ N ), VCN sch ( • N ) satisfies condition (c) of Definition 3.11, (i), in the case where we take “i” to be n. Thus, we conclude [cf. the fact that ◦ N n−1 , • N n−1 are VCN-complete] that ◦ N , • N are VCN-complete. This completes the proof of Claim 3.14.A, hence also the proofs of assertions (iii), (iv). COMBINATORIAL ANABELIAN TOPICS III 77 Finally, we verify assertion (v). For each  ∈ {◦, •} and each † i ∈ {1, . . . , n}, write  H i/i−1 ,  H i/i−1 for the respective subquotients † of  H,  H determined by the subquotient  Π i/i−1 of  Π n ; (  H i/i−1 ) {l} , † † (  H i/i−1 ) {l} for the respective maximal pro-l quotients of  H i/i−1 ,  H i/i−1 [cf. Definition 3.10, (c)]. Then let us observe that it follows immedi- ately from [CmbGC], Proposition 1.2, (ii), together with the various definitions involved that, for each  ∈ {◦, •} and each i ∈ {1, . . . , n}, every VCN-subgroup of (  H i/i−1 ) {l} [i.e., discussed as in Definition 3.12, (iii), (iv)] may be obtained as the commensurator of the image of a † † VCN-subgroup of (  H i/i−1 ) {l} [by the homomorphism (  H i/i−1 ) {l} → † (  H i/i−1 ) {l} determined by the natural inclusion  H ⊆  H]. More- over, one also verifies easily that every proper closed subgroup of (  H i/i−1 ) {l} obtained as the commensurator of the image of a VCN-subgroup of † (  H i/i−1 ) {l} is a VCN-subgroup of (  H i/i−1 ) {l} . Assertion (v) now fol- lows formally. This completes the proof of Lemma 3.14.  Definition 3.15. In the notation of Definition 3.12, write (F i ) i∈{1,··· ,n} ∈ VCN gp (  H) for the VCN-chain of  H associated to y  ∈ VCN sch (  H) [cf. Definition 3.12, (iv)]. Now since (  H) {l} ⊆ (  Π n ) ∗ [cf. the notation of condition (b) of Definition 3.10] is open, and (  Π n ) ∗ is topologically finitely generated, slim [cf. Proposition 2.3, (i)] and almost pro-l, there exist an open subgroup  J ⊆ I  K of I  K and a homomorphism  ρ :  J −→ Out((  H) {l} ) that • is compatible [in the evident sense] with the homomorphism  J → Out((  Π n ) ∗ ) induced [cf. condition (a) of Definition 3.10] by  ρ n : I  K → Out(  Π n ), • induces, for each i ∈ {1, · · · , n}, a homomorphism  J −→ Out((  H i ) {l} ) — relative to the natural surjection (  H) {l}  (  H i ) {l} — and, moreover, • factors through the maximal pro-l quotient (  J) {l} of  J, which [as is easily verified] is isomorphic to Z l as an abstract profinite group. def Write I y  0 = (  J) {l} . Then, for i ∈ {1, · · · , n}, we define closed sub- groups I y  i ⊆  ρ H i | y  i−1 ⊆  ρ def out H i = (  H i ) {l}  (  J) {l} 78 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [cf. the discussion entitled “Topological groups” in [CbTpI], §0] as fol- lows [inductively on i]: (i) Set  ρ ρ H 1 | y  0 =  H 1 , I y  1 = Z  H ρ 1 | y  (F 1 ). def def 0 (ii) Suppose that n ≥ i ≥ 2. Then, by the induction hypothesis, we have already constructed closed subgroups I y  i−1 ⊆  ρ H i−1 | y  i−2 ⊆  ρ H i−1 , hence also a natural outer representation ρ I y  i−1 →  H i−1 → Out((  H i/i−1 ) {l} ) — where the second arrow is the natural outer representation arising from the exact sequence of profinite groups ρ ρ 1 −→ (  H i/i−1 ) {l} −→  H i −→  H i−1 −→ 1. Then we set  ρ out H i | y  i−1 = (  H i/i−1 ) {l}  I y  i−1 , I y  i = Z  H ρi | y  def def i−1 (F i ). Remark 3.15.1. In the situation of Definition 3.15, it follows imme- diately from the definition of I y  i [cf. also [CmbGC], Remark 1.1.3; [CmbGC], Proposition 1.2, (ii)] that I y  i is isomorphic to a profinite group of the form Z ⊕j l , where j is a positive integer ≤ i + 1. Proposition 3.16 (Graph-admissible isomorphisms). In the no- ∼ tation of Definition 3.1, let α : ◦ Π n → • Π n be an isomorphism of profi- nite groups [so ◦ Σ = • Σ — cf., e.g., the proof of [CbTpI], Proposition 1.5, (i)] and l ∈ ◦ Σ = • Σ such that l ∈ { ◦ p, • p}. Then the following hold: (i) If ◦ p ∈ ◦ Σ and • p ∈ • Σ, then suppose that α is PC-admissible [cf. [CbTpI], Definition 1.4, (ii)]. If α is SAF-admissible [cf. Definition 3.13, (i)] and {l}-I-admissible [cf. Defini- tion 3.8, (i)], then α is {l}-G-admissible [cf. Definition 3.13, (ii)]. (ii) Suppose that α is {l}-G-admissible. Then there exists an algorithm, which is functorial with respect to α, for con- structing an isomorphism of topological groups ∼ • tp α tp : ◦ Π tp n −→ Π n ∼ such that the isomorphism ◦ Π n → • Π n induced by α tp [cf. Proposition 3.3, (i)] coincides with α. COMBINATORIAL ANABELIAN TOPICS III 79 Proof. First, we verify assertion (i). Let ◦ J ⊆ ◦ Π n be an open sub- group of ◦ Π n . Then it follows from Lemma 3.14, (i), (ii), (iii), that there exist an open subgroup ◦ H ⊆ ◦ Π n of ◦ Π n of l-polystable type [cf. Definition 3.10] and an ◦ H-l-system ◦ H = { ◦ H λ } λ∈Λ [cf. Defini- def tion 3.11, (ii)] such that ◦ H ⊆ ◦ J, • H = α( ◦ H) is of l-polystable type, def and • H = { • H λ = α( ◦ H λ )} λ∈Λ is an • H-l-system. Now it follows im- mediately from the various definitions involved that, to complete the verification of assertion (i), it suffices to verify the following assertion: Claim 3.16.A: For each i ∈ {1, · · · , n}, the isomor- ∼ phism ◦ H i → • H i [cf. the notation of Definition 3.10] determined by α induces a bijection ∼ VCN gp ( ◦ H i ) −→ VCN gp ( • H i ) [cf. Definitions 3.11, (iii); 3.12, (iv)]. We verify Claim 3.16.A by induction on i. If i = 1, then Claim 3.16.A follows immediately from the equivalence (a) ⇔ (b ∃ ) of Theorem 3.9, together with Remark 3.13.1, (i). Now suppose that i ≥ 2, and that the induction hypothesis is in force. Then it follows immediately from the induction hypothesis that, for each j ∈ {1, · · · , i − 1}, the isomorphism ∼ ◦ H j → • H j determined by α induces a bijection ∼ VCN gp ( ◦ H j ) −→ VCN gp ( • H j ). Let ◦ y  i−1 ∈ VCN sch ( ◦ H i−1 ), • y  i−1 ∈ VCN sch ( • H i−1 ) [cf. Definition 3.11, (iii)] be elements that correspond via the above bijection, relative to the ◦-, •-versions of the displayed bijection of Definition 3.12, (iv). Now since α is {l}-I-admissible, for  ∈ {◦, •}, there exist an open subgroup  J ⊆ I  K of I  K and an outer representation  ρ :  J → Out((  H) {l} ) as in Definition 3.15 such that ◦ ρ is compatible, relative to α, with • ρ. Thus, it follows immediately from the various definitions ∼ involved that the isomorphism ◦ H i → • H i determined by α induces an isomorphism of profinite groups ◦ ∼ H ρi | ◦ y  i−1 −→ • H ρi | • y  i−1 ∼ that lies over an isomorphism β : I ◦ y  i−1 → I • y  i−1 [cf. Definition 3.15]. In particular, we obtain a commutative diagram of profinite groups I ◦ y  i−1 −−−→ Out(( ◦ H i/i−1 ) {l} ) ⏐ ⏐ ⏐ ⏐ β   I • y  i−1 −−−→ Out(( • H i/i−1 ) {l} ) — where the right-hand vertical arrow is the isomorphism induced by α. Moreover, one verifies immediately from the various definitions involved [cf. also Remark 3.15.1] that, for each  ∈ {◦, •}, the pos- itive definite profinite Dehn multi-twists [cf. [CbTpI], Definition 4.4; 80 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [CbTpI], Definition 5.8, (iii)] in the image of the composite ∼ I  y  i−1 −→ Out((  H i/i−1 ) {l} ) ←− Out(Π G i,  y  i−1 ) — where the second arrow is the isomorphism induced by the isomor- phism of Definition 3.12, (iii) — form a dense subset of this image [cf. [CbTpI], Lemma 5.4, (i), (ii), (iii); [CbTpI], Proposition 5.6, (ii)]. In particular, it follows immediately [cf. the easily verified elementary fact that no dense subset of a nonzero finitely generated free Z l -module is contained in a finite union of proper Z l -submodules of the given finitely generated free Z l -module] that there exists an element ◦ γ ∈ I ◦ y  i−1 such def that if we write • γ = β( ◦ γ) ∈ I • y  i−1 , then, for  = ◦ (respectively,  = •), the image of  γ via the composite of the above display is a positive definite profinite Dehn multi-twist (respectively, nondegenerate profinite Dehn multi-twist [cf. [CbTpI], Definition 4.4; [CbTpI], Defi- nition 5.8, (ii)]). Thus, it follows immediately from [CbTpII], Theorem 1.9, (ii), together with the equivalences of [CbTpI], Corollary 5.9, (ii), (iii), that the isomorphism ∼ ∼ ∼ α i/i−1 : Π G i, ◦ y  i−1 −→ ( ◦ H i/i−1 ) {l} −→ ( • H i/i−1 ) {l} ←− Π G i, • y  i−1 induced by α is group-theoretically verticial, hence also group-theoretically nodal. Next, let us observe that it follows from the fact that α i/i−1 is group- theoretically verticial [hence also group-theoretically nodal], together with our assumption concerning PC-admissibility, that if ◦ p ∈ ◦ Σ and • p ∈ • Σ, then [cf. [CmbGC], Proposition 1.5, (ii)] α i/i−1 is graphic. On the other hand, if either ◦ p ∈ ◦ Σ or • p ∈ • Σ, then it follows from Propo- sition 3.6, (iii) [applied to “(c ∃ )” — cf. Remark 3.13.1, (i)], together with Claim 3.16.A in the case where i = 1, that ◦ p = • p ∈ ◦ Σ = • Σ. In particular, if either ◦ p ∈ ◦ Σ or • p ∈ • Σ, then, by allowing the open sub- group “ ◦ H” of ◦ Π n to vary and applying the group-theoretic nodality of the resulting isomorphisms “α i/i−1 ”, one concludes from the “existence of irreducible components that collapse to arbitrary cusps” [cf. the proof of “observation (iv)” given in the proof of [SemiAn], Corollary 3.11; [SemiAn], Remark 3.11.1; [AbsTpII], Corollary 2.11; [AbsTpII], Remark 2.11.1, (i)] that α i/i−1 is group-theoretically cuspidal, hence also [cf. [CmbGC], Proposition 1.5, (ii)] graphic. Thus, by allowing ◦ y  i−1 , • y  i−1 to vary, we conclude immediately from the various definitions involved that Claim 3.16.A holds. This completes the proof of Claim 3.16.A, hence also of assertion (i). Next, we verify assertion (ii). The theory of [Brk] yields • a functorial homotopy [indeed, a proper strong deformation retraction!] between the skeleton of a polystable fibration [cf. COMBINATORIAL ANABELIAN TOPICS III 81 [Brk], Definitions 1.2, 1.3] over the ring of integers of a com- plete nonarchimedean field and the analytic space associated to the polystable fibration [cf. [Brk], Theorem 8.1], as well as • a functorial homeomorphism between the skeleton of a polystable fibration over the ring of integers of a complete nonarchimedean field and the geometric realization of a certain polysimplicial set associated to the special fiber of the polystable fibration [cf. [Brk], Theorem 8.2]. In particular, the theory of [Brk] gives rise to a functorial homotopy between the analytic space associated to a polystable fibration over the ring of integers of a complete nonar- chimedean field and the geometric realization of a cer- tain polysimplicial set associated to the special fiber of the polystable fibration. Here, we recall further that this polysimplicial set is completely de- termined by the set of strata of the special fiber, together with the specialization/generization relations between these strata [cf. the dis- cussion surrounding [Brk], Proposition 2.1, and its proof; [Brk], Lemma 3.13; [Brk], Lemma 6.7]. Next, let us observe that the various bijections ∼ ∼ ∼ VCN sch ( ◦ H) −→ VCN gp ( ◦ H) −→ VCN gp ( • H) ←− VCN sch ( • H) [cf. Definitions 3.12, (iv); 3.13, (ii)] induced by an {l}-G-admissible iso- ∼ morphism ◦ Π n → • Π n induce bijections between the respective sets of strata of the special fibers of ◦ Y n , • Y n [cf. the notation of condition (e) of Remark 3.10.1, (i)], which, in light of the group-theoretic descriptions of specialization/generization relations given in [CbTpI], Proposition 2.9, (i) [cf. also [CbTpI], Proposition 5.6, (iii), (iv)], are [easily seen to be] compatible with these specialization/generization relations. In par- log ticular, since each log scheme  Y n gives rise to a polystable fibration as in the above discussion of [Brk] [cf. condition (e) of Remark 3.10.1, (i)], we thus conclude, in light of the theory of [Brk], from the defini- tion of the tempered fundamental group given in [André], §4.2 [cf. also the discussion of Definition 3.1, (ii), of the present paper], that any ∼ {l}-G-admissible isomorphism ◦ Π n → • Π n determines an isomorphism ◦ ∼ • tp Π tp n −→ Π n between the respective tempered fundamental groups, which gives back ∼ the original isomorphism ◦ Π n → • Π n upon passing to the respective ◦ Σ = • Σ-completions [cf. Proposition 3.3, (i)]. This completes the proof of assertion (ii).  82 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Theorem 3.17 (Metric-, inertia-admissible outomorphisms of fundamental groups). Let n be a positive integer; (g, r) a pair of nonnegative integers such that 2g − 2 + r > 0; p a prime number; Σ a nonempty set of prime numbers such that Σ  = {p}, and, moreover, if n ≥ 2, then Σ is either equal to the set of all prime numbers or of cardinality one; R a mixed characteristic complete discrete valuation ring of residue characteristic p whose residue field is separably closed; K the field of fractions of R; K an algebraic closure of K; log X K a smooth log curve of type (g, r) over K. Write (X K ) log n for the n-th log configuration space [cf. the discussion entitled def log log over K; (X K ) log “Curves” in [CbTpII], §0] of X K n = (X K ) n × K K; def Σ Π n = π 1 ((X K ) log n ) for the maximal pro-Σ quotient of the log fundamental group of (X K ) log n ; def ρ n : I K = Gal(K/K) −→ Out(Π n ) log for the natural outer pro-Σ Galois action associated to (X K ) log n ; (Spec R) for the log scheme obtained by equipping Spec R with the log structure determined by the closed point of Spec R. Then the following hold: (i) Let l ∈ Σ be such that l  = p. Then we have equalities and an inclusion Out(Π 1 ) M = Out I (Π 1 ) ∩ Out C (Π 1 ) = Out {l}-I (Π 1 ) ∩ Out C (Π 1 ) ⊆ Out(Π 1 ) G [cf. Definitions 3.7, (i), (ii); 3.8, (iii)]. If, moreover, p ∈ Σ, then we have equalities and inclusions Out(Π 1 ) M = Out I (Π 1 ) = Out {l}-I (Π 1 ) ⊆ Out(Π 1 ) G ⊆ Out(Π 1 ). (ii) Let l ∈ Σ be such that l  = p. Then we have equalities and inclusions Out FC (Π n ) M = Out FCI (Π n ) = Out FC (Π n ) I = Out FC {l}-I (Π n ) = Out FC (Π n ) {l}-I ⊆ Out G (Π n ) ⊆ Out {l}-G (Π n ), Out FC (Π n ) M ⊆ Out FI (Π n ) ⊆ Out F {l}-I (Π n ) ∩ ∩ ∩ F F F M I ⊆ Out (Π n ) ⊆ Out (Π n ) {l}-I Out (Π n ) [cf. Definitions 3.7, (iii); 3.8, (iii), (iv); 3.13, (iv)]. Moreover, the following hold: COMBINATORIAL ANABELIAN TOPICS III 83 (ii-a) If p ∈ Σ, then we have: Out FC (Π n ) M = Out FI (Π n ) = Out F {l}-I (Π n ), Out F (Π n ) M = Out F (Π n ) I = Out F (Π n ) {l}-I . (ii-b) If n  = 1, then we have: Out FI (Π n ) = Out F {l}-I (Π n ), Out F (Π n ) M = Out F (Π n ) I = Out F (Π n ) {l}-I . (ii-c) If n  = 2, (r, n)  = (0, 3), and either p ∈ Σ or n  = 1, then we have: Out F (Π n ) M = Out FI (Π n ) = Out F (Π n ) I = Out F {l}-I (Π n ) = Out F (Π n ) {l}-I FC = Out (Π n ) M = Out FCI (Π n ) = Out FC (Π n ) I = Out FC {l}-I (Π n ) = Out FC (Π n ) {l}-I . log (iii) Suppose that p ∈ Σ, and that X K extends to a stable log log curve over (Spec R) . Let l ∈ Σ. Write ρ n (I K )[l] ⊆ ρ n (I K ) for the maximal pro-l subgroup of the [necessarily pro-cyclic — cf. the injectivity portion of [NodNon], Theorem B; the dis- cussion of [CbTpI], Definition 5.3] image ρ n (I K ). Then the normalizers of ρ n (I K ), ρ n (I K )[l] in Out F (Π n ) satisfy the fol- lowing equalities: (iii-a) If (r, n)  = (0, 2), then Out FI (Π n ) = Out F (Π n ) I = N Out F (Π n ) (ρ n (I K )), Out F {l}-I (Π n ) = Out F (Π n ) {l}-I = N Out F (Π n ) (ρ n (I K )[l]). (iii-b) For arbitrary r ≥ 0, n ≥ 1, Out FI (Π n ) = N Out F (Π n ) (ρ n (I K )), Out F {l}-I (Π n ) = N Out F (Π n ) (ρ n (I K )[l]). (iv) Let l ∈ Σ be such that l  = p. Then the subgroups Out(Π 1 ) M , Out I (Π 1 ), Out {l}-I (Π 1 ), Out(Π 1 ) G of Out(Π 1 ) are closed in Out(Π 1 ). Moreover, the subgroups Out F (Π n ) M , Out FI (Π n ), Out F (Π n ) I , F {l}-I (Π n ), Out F (Π n ) {l}-I , Out Out FC (Π n ) M , Out FCI (Π n ), Out FC (Π n ) I , Out FC {l}-I (Π n ), Out FC (Π n ) {l}-I , Out {l}-G (Π n ) Out G (Π n ), of Out(Π n ) are closed in Out(Π n ). In particular, these sub- groups are compact. 84 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (v) Let l ∈ Σ be such that l  = p. Suppose, moreover, that the log arises, via base-change, from a smooth smooth log curve X K log curve over a complete discrete valuation field whose residue field is finitely generated over a finite field. Then the closed subgroups Out G (Π n ), Out {l}-G (Π n ) ⊆ Out F (Π n ) [cf. (iv); Remark 3.13.1, (ii)] are commensurably terminal in Out F (Π n ). Moreover, we have an inclusion C Out F (Π n ) (Out FC (Π n ) M ) ⊆ Out G (Π n ). (vi) The natural homomorphism Out FC (Π n+1 ) M −→ Out FC (Π n ) M (respectively, Out F (Π n+1 ) M −→ Out F (Π n ) M ) log induced by the projection (X K ) log n+1 → (X K ) n obtained by for- getting any one of the n + 1 factors is injective (respectively, injective if (r, n)  = (0, 1)). If, moreover, either n ≥ 4 or n ≥ 3 and r  = 0, then this homomorphism is bijective (respectively, bijective). Proof. Assertion (i) follows immediately from Theorem 3.9. Next, we verify assertion (ii). First, we claim that the following assertion holds: Claim 3.17.A: We have equalities Out FC (Π n ) I = Out FCI (Π n ); Out FC (Π n ) {l}-I = Out FC {l}-I (Π n ). Indeed, this follows immediately — in light of the definition of I- admissibility, {l}-I-admissibility [cf. Definition 3.8] — from Proposi- tion 2.3, (ii), and Corollary 2.10 [when Σ = Primes]; the injectivity portion of [NodNon], Theorem B [when Σ = {l}]. This completes the proof of Claim 3.17.A. Next, we claim that the following assertion holds: Claim 3.17.B: We have equalities Out FC (Π n ) M = Out FC (Π n ) I = Out FC (Π n ) {l}-I . Indeed, this follows immediately from assertion (i), together with the various definitions involved. This completes the proof of Claim 3.17.B. Next, we claim that the following assertion holds: Claim 3.17.C: We have equalities and an inclusion Out FC (Π n ) M = Out FC (Π n ) I = Out FC (Π n ) {l}-I = Out FCI (Π n ) = Out FC {l}-I (Π n ) ⊆ Out G (Π n ). COMBINATORIAL ANABELIAN TOPICS III 85 Indeed, the first four equalities follow from Claims 3.17.A, 3.17.B. On the other hand, the final inclusion follows immediately from Proposi- tion 3.16, (i) [cf. also the final portion of Definition 3.13, (i)]. This completes the proof of Claim 3.17.C. Next, we claim that the following assertion holds: Claim 3.17.D: We have inclusions Out FC (Π n ) M ⊆ Out FI (Π n ) ⊆ Out F {l}-I (Π n ) ∩ ∩ ∩ F F F M I Out (Π n ) ⊆ Out (Π n ) ⊆ Out (Π n ) {l}-I . Indeed, let us observe that the left-hand upper inclusion follows imme- diately from Claim 3.17.C. Next, let us observe that the left-hand lower inclusion follows immediately from assertion (i). On the other hand, the remaining inclusions follow immediately from the various defini- tions involved. This completes the proof of Claim 3.17.D. The various equalities and inclusions of assertion (ii) that precede assertion (ii-a) all follow from Claims 3.17.C, 3.17.D. Next, we consider assertion (ii-a). It follows immediately from Propo- sition 3.16, (i), that the inclusion Out F {l}-I (Π n ) ⊆ Out {l}-G (Π n ) holds. In particular, it follows from Remark 3.13.1, (ii), that the inclusion Out F {l}-I (Π n ) ⊆ Out FC (Π n ), hence also the equality Out F {l}-I (Π n ) = Out FC {l}-I (Π n ), holds. Thus, the first two equalities of assertion (ii-a) follow immediately from Claims 3.17.C, 3.17.D. On the other hand, the final two equalities of assertion (ii-a) follow immediately from the final portion of assertion (i). This completes the proof of assertion (ii-a). Next, we consider assertion (ii-b). If p ∈ Σ, then assertion (ii-b) follows from assertion (ii-a). Thus, we may assume without loss of generality that p ∈ Σ. Then since [by assumption!] Σ = {l}, the first equality of assertion (ii-b) follows immediately from the various definitions involved. On the other hand, the final two equalities follow immediately from assertion (i), together with [CbTpI], Theorem A, (ii). This completes the proof of assertion (ii-b). Next, we consider assertion (ii-c). If n ≥ 3 and (r, n)  = (0, 3), then assertion (ii-c) follows immediately from [CbTpII], Theorem A, (ii), together with Claim 3.17.C. On the other hand, if p ∈ Σ and n = 1, then assertion (ii-c) follows immediately from the final portion of assertion (i), together with Claim 3.17.C [cf. also Remark 3.13.1, (ii)]. This completes the proof of assertion (ii-c), hence also of assertion (ii). Next, we verify assertion (iii). First, we claim that the following assertion holds: Claim 3.17.E: We have an equality Out {l}-I (Π 1 ) = N Out(Π 1 ) (ρ 1 (I K )[l]). Indeed, let us first observe that since p ∈ Σ, we have a natural outer ∼ isomorphism Π 1 → Π G Π1 [Σ] [cf. Remark 3.5.1]. Next, let us observe 86 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI log that, in light of our assumption that X K extends to a stable log curve over (Spec R) log , it follows from [CbTpI], Definition 5.3, (i), that the image of ρ 1 is contained in ∼ Dehn(G Π 1 [Σ]) ⊆ Out(Π G Π1 [Σ] ) ← Out(Π 1 ) [cf. [CbTpI], Definition 4.4] and, moreover, is pro-cyclic. Next, let us observe that it follows immediately from the definition of {l}-I- admissibility that N Out(Π 1 ) (ρ 1 (I K )[l]) ⊆ Out {l}-I (Π 1 ). Thus, to com- plete the verification of Claim 3.17.E, it suffices to verify that we have an inclusion Out {l}-I (Π 1 ) ⊆ N Out(Π 1 ) (ρ 1 (I K )[l]). Let α ∈ Out {l}-I (Π 1 ) and H ⊆ Π 1 a characteristic open subgroup of Π 1 . Write Π 1  Π ∗ 1 for the maximal almost pro-l quotient of Π 1 with respect to H [cf. Definition 1.1]; Π G Π1 [Σ]  Π ∗G Π [Σ] for the [necessarily 1 maximal almost pro-l] quotient of Π G Π1 [Σ] corresponding to Π 1  Π ∗ 1 ∼ [relative to the above natural outer isomorphism Π 1 → Π G Π1 [Σ] ]. Then it follows immediately [in light of the well-known simple structure of pro-cyclic profinite groups] from the definition of {l}-I-admissibility that there exists an open subgroup J ⊆ I K such that the image of ρ 1 (J) in Out(Π ∗ 1 ) is normalized by the outomorphism α ∗ ∈ Out(Π ∗ 1 ) determined by α ∈ Out(Π 1 ). On the other hand, it follows immediately from the above discussion that the outer representation ∼ ρ 1 (J) −→ Out(Π ∗ 1 ) −→ Out(Π ∗G Π [Σ] ) 1 is of PIPSC-type [cf. Definition 1.6, (iv)]. Thus, it follows from The- orem 1.11, (ii), that α ∗ ∈ Out(Π ∗ 1 ) is group-theoretical verticial [cf. Definition 1.6, (ii)]. In particular, by allowing H to vary, we con- clude that α ∈ Out(Π 1 ) is group-theoretical verticial. Thus, it follows immediately from the definition of a profinite Dehn multi-twist that ∼ α ∈ Out(Π 1 ) normalizes Dehn(G Π 1 [Σ]) ⊆ Out(Π G Π1 [Σ] ) ← Out(Π 1 ), hence also [cf. [CbTpI], Theorem 4.8, (iv)] the maximal pro-l subgroup Dehn(G Π 1 [Σ])[l] of Dehn(G Π 1 [Σ]). On the other hand, one verifies im- mediately again from [CbTpI], Theorem 4.8, (iv), that Dehn(G Π 1 [Σ])[l] is a free Z l -module of finite rank, and that the composite Dehn(G Π 1 [Σ])[l] → Out(Π 1 ) → Out(Π ∗ 1 ) is injective. Thus, since some open subgroup of the maximal pro-l sub- group of the image of I K in Out(Π ∗ 1 ) is normalized by α ∗ ∈ Out(Π ∗ 1 ) [cf. the above discussion concerning “J”!], one verifies immediately [from well-known elementary properties of free Z l -modules of finite rank] that α ∈ N Out(Π 1 ) (ρ 1 (I K )[l]). This completes the proof of Claim 3.17.E. Now let us observe that one verifies easily [cf. also the discussion of the inclusion “N Out(Π 1 ) (ρ 1 (I K )[l]) ⊆ Out {l}-I (Π 1 )” in the proof of Claim 3.17.E] that the inclusions N Out F (Π n ) (ρ n (I K )) ⊆ Out FI (Π n ) ⊆ Out F (Π n ) I , COMBINATORIAL ANABELIAN TOPICS III 87 N Out F (Π n ) (ρ n (I K )[l]) ⊆ Out F {l}-I (Π n ) ⊆ Out F (Π n ) {l}-I hold. In particular, assertion (iii-a) follows immediately from Claim 3.17.E, together with the injectivity portion of [CbTpII], Theorem A, (i) [cf. also [CbTpI], Theorem A, (ii); [NodNon], Theorem B, in the case where r = 0]. Thus, to complete the proof of assertion (iii), it suffices to verify the two equalities of assertion (iii-b) in the case where (r, n) = (0, 2). Suppose that (r, n) = (0, 2), hence that Σ = {l}. Then one verifies easily that, to complete the proof of assertion (iii), it suffices to verify that Out F {l}-I (Π n ) ⊆ N Out F (Π n ) (ρ n (I K )[l]). Thus, let α  ∈ Aut(Π n ) be a lifting of an element α ∈ Out F {l}-I (Π n ). Then let us observe that it follows immediately from Claim 3.17.E that out α  induces an automorphism β  of the extension group Π 1  ρ 1 (I K )[l] [i.e., arising from the outer representation of IPSC-type ρ 1 (I K ) → Out(Π 1 ) implicit in the discussion surrounding Claim 3.17.E above], whose restriction to Π 1 is G-admissible [cf. assertion (i)]. In particular, out it follows that β  maps verticial inertia groups of Π 1  ρ 1 (I K )[l] [each of which surjects onto ρ 1 (I K )[l] — cf. [NodNon], Definition 2.2, (i); [NodNon], Definition 2.4, (ii); [NodNon], Remark 2.4.2] to verticial in- out ertia groups of Π 1  ρ 1 (I K )[l]. Moreover, let us observe that it follows immediately from the fact that α ∈ Out F {l}-I (Π n ) that β  is compat- ible with the natural outer representations of suitable open subgroups out of such verticial inertia groups of Π 1  ρ 1 (I K )[l] on Π 2/1 [relative to the action of α  on Π 2/1 ]. Thus, since the natural outer representation out of such a verticial inertia group of Π 1  ρ 1 (I K )[l] on Π 2/1 is [eas- ily verified to be] an outer representation of IPSC-type, one concludes from a similar argument to the [final portion of the] argument applied above to verify Claim 3.17.E that β  is compatible with these natural out outer representations of verticial inertia groups of Π 1  ρ 1 (I K )[l] on  on Π 2/1 ]. Now it follows formally Π 2/1 [relative to the action of α that α ∈ N Out F (Π n ) (ρ n (I K )[l]), as desired. This completes the proof of assertion of (iii). Next, we verify assertion (iv). The closedness of Out(Π 1 ) G in Out(Π 1 ) follows immediately from condition (c ∀ ) of Proposition 3.6 [cf. Propo- sition 3.6, (ii)]. Thus, the closedness of Out(Π 1 ) M in Out(Π 1 ) follows from the easily verified fact that Out M (Π 1 ) is closed in Out(Π 1 ) G . The fact that the subgroup Out {l}-I (Π 1 ), hence also Out I (Π 1 ), is closed in Out(Π 1 ) may be verified as follows: If p ∈ Σ, then the closedness in question follows from the closedness of Out(Π 1 ) M [verified above], to- gether with the final portion of assertion (i). On the other hand, if p ∈ Σ, then the closedness in question follows immediately from as- sertion (iii). This completes the proof of the closedness of Out {l}-I (Π 1 ) and Out I (Π 1 ) in Out(Π 1 ). 88 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI The closedness of Out FC (Π n ) M , Out FC (Π n ) I , Out FC (Π n ) {l}-I , Out F (Π n ) M , Out F (Π n ) I , Out F (Π n ) {l}-I in Out(Π n ) follows immediately from the various definitions involved, together with the closedness of Out(Π 1 ) M , Out I (Π 1 ), and Out {l}-I (Π 1 ) in Out(Π 1 ) [verified above]. The closedness of Out FCI (Π n ), Out FC {l}-I (Π n ) in Out(Π n ) follow from the closedness of Out FC (Π n ) M in Out(Π n ) [ver- ified above], together with the equalities at the beginning of assertion (ii). Next, we verify the closedness of Out G (Π n ), Out {l}-G (Π n ) in Out(Π n ). Let us first observe that it is immediate that, to verify the desired closedness, it suffices to verify the closedness of Out {l}-G (Π n ) in Out(Π n ). Next, to verify the closedness of Out {l}-G (Π n ) in Out(Π n ), let (α ξ ) ξ≥1 be a sequence [indexed by the positive integers] of elements ∈ Out {l}-G (Π n ) that converges to an element α ∞ ∈ Out(Π n ). Then since [one verifies easily that] the subgroup of Out(Π n ) consisting of SAF-admissible outomorphisms is closed, α ∞ is SAF-admissible. Next, to verify that α ∞ satisfies the condition of Definition 3.13, (ii), let us fix an open subgroup J ⊆ Π n of Π n . Now we define open subgroups H ξ ⊆ J ξ ⊆ Π n of Π n inductively on ξ as follows: def • Set J 1 = J. • Suppose that J ξ ⊆ Π n has already been defined. Then since α ξ is {l}-G-admissible, there exists an open subgroup H ⊆ ◦ Π n of ◦ Π n of l-polystable type such that H ⊆ J ξ , and, moreover, α ξ satisfies the condition of Definition 3.13, (ii), in the case where we take the “( ◦ J, ◦ H)” of Definition 3.13, (ii), to be (J ξ , H). def Then define H ξ = H. • Suppose that ξ ≥ 2, and that H ξ−1 ⊆ Π n has already been def defined. Then set J ξ = H ξ−1 . Then it follows immediately from Lemma 3.14, (iii), (v), that α ∞ sat- isfies the condition of Definition 3.13, (ii), in the case where we take the “( ◦ J, ◦ H)” of Definition 3.13, (ii), to be (J = J 1 , H = H 1 ). In particular, the SAF-admissible outomorphism α ∞ is {l}-G-admissible, as desired. This completes the proof of the closedness of Out {l}-G (Π n ) in Out(Π n ). The fact that the subgroup Out F {l}-I (Π n ), hence also Out FI (Π n ), is closed in Out(Π n ) may be verified as follows: If n = 1, then the closed- ness in question has already been verified. If p ∈ Σ, then the closedness in question follows from the closedness of Out FC (Π n ) M [verified above], COMBINATORIAL ANABELIAN TOPICS III 89 together with assertion (ii-a). On the other hand, if p ∈ Σ, then the closedness in question follows from assertion (iii-b). This completes the proof of the closedness of Out F {l}-I (Π n ), Out FI (Π n ) in Out(Π n ), hence also of assertion (iv). Next, we verify assertion (v). Let α ∈ C Out F (Π n ) (Out G (Π n )) (re- spectively, C Out F (Π n ) (Out {l}-G (Π n )); C Out F (Π n ) (Out FC (Π n ) M )) and α  ∈ F Aut (Π n ) a lifting of α. Now observe that to complete the verifi- cation of assertion (v), it suffices to verify that α ∈ Out {l}-G (Π n ). To this end, let J ⊆ Π n be an open subgroup of Π n . Then it fol- lows from Lemma 3.14, (i), (iii), that there exist an open subgroup H ⊆ J ⊆ Π n of Π n of l-polystable type [cf. Definition 3.10] and an H-l-system H = {H λ } λ∈Λ [cf. Definition 3.11, (ii)]. Note that it fol- lows from condition (a) of Definition 3.10 that the subgroups H, H λ of Π n are stabilized by α  . Then it follows immediately from the various definitions involved that, to complete the verification of the fact that α ∈ Out {l}-G (Π n ), it suffices to verify the following assertion: Claim 3.17.F: For each i ∈ {0, · · · , n}, the outomor- phism of the image H i of H in Π i determined by α induces a bijection ∼ VCN gp (H i ) −→ VCN gp (H i ) [cf. Definition 3.11, (iii); Definition 3.12, (iv)] — where, def def for convenience, we set Π 0 = {1}, VCN gp (H 0 ) = {Π 0 }, and we write (H λ ) i for the image of H λ in Π i def and H i = {(H λ ) i } λ∈Λ . We verify Claim 3.17.F by induction on i. If i = 0, then Claim 3.17.F is immediate. Now suppose that i ≥ 1, and that the induction hypothesis is in force. Then it follows immediately from the induction hypothesis that, for each j ∈ {0, · · · , i − 1}, the outomorphism of H j determined by α induces a bijection ∼ VCN gp (H j ) −→ VCN gp (H j ). Let y  , y   ∈ VCN sch (H i−1 ) [cf. Definition 3.11, (iii)] be elements that correspond via the bijection obtained by conjugating the above bijec- tion by the displayed bijection of Definition 3.12, (iv). Next, let us observe that since α ∈ C Out F (Π n ) (Out G (Π n )) (respec- tively, C Out F (Π n ) (Out {l}-G (Π n )); C Out F (Π n ) (Out FC (Π n ) M )), there exist open subgroups N 1 and N 2 of Out G (Π n ) (respectively, Out {l}-G (Π n );  extends Out FC (Π n ) M ) such that the automorphism of H i induced by α to an isomorphism of profinite groups [cf. assertion (iv)] out ∼ out H i  N 1 −→ H i  N 2 90 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [cf. the discussion entitled “Topological groups” in [CbTpI], §0] that lies ∼ over an isomorphism of profinite groups N 1 → N 2 . In particular, by out considering the respective outer actions [by conjugation] of H i−1  N 1 , out H i−1  N 2 on the maximal pro-l quotient (H i/i−1 ) {l} of the kernel def H i/i−1 = Ker(H i  H i−1 ) [cf. the notation of Remark 3.10.1, (i)], we obtain a commutative diagram of profinite groups out ∼ out ∼ H i−1  N 1 −−−→ Out((H i/i−1 ) {l} ) ←−−− Out(Π G i, y ) ⏐ ⏐ ⏐ ⏐  ⏐  ⏐     H i−1  N 2 −−−→ Out((H i/i−1 ) {l} ) ←−−− Out(Π G i, y  ) — where the left-hand vertical arrow is the isomorphism induced by the isomorphism of profinite groups discussed above; the central verti- cal arrow is the automorphism induced by α  ; the right-hand horizontal arrows are the isomorphisms induced by the y  -, y   -versions of the iso- morphism of Definition 3.12, (iii); the right-hand vertical arrow is the isomorphism induced by the composite ∼ ∼ ∼ α  y  , y  : Π G i, y −→ (H i/i−1 ) {l} −→ (H i/i−1 ) {l} ←− Π G i, y  ∼ of the isomorphism Π G i, y → (H i/i−1 ) {l} of Definition 3.12, (iii), the automorphism of (H i/i−1 ) {l} determined by α  , and the isomorphism {l} ∼ (H i/i−1 ) ← Π G i, y  of Definition 3.12, (iii). Now let us recall that we have assumed that the smooth log curve log X K arises, via base-change, from a smooth log curve over a complete discrete valuation field whose residue field is finitely generated over a finite field. In particular, one verifies immediately from the openness of N 1 , N 2 in Out G (Π n ) (respectively, Out {l}-G (Π n ); Out FC (Π n ) M = Out FC (Π n ) I ⊆ Out {l}-G (Π n ) [cf. assertion (ii)]) that the composite horizontal arrows of the above commutative diagram factor through Aut(G i, y ), Aut(G i, y  ), respectively, and, moreover, are l-graphically full [i.e., in the sense of [CmbGC], Definition 2.3, (iii)] — cf. the argu- ment applied in the proof of [CmbGC], Proposition 2.4, (v). Thus, it follows from [CmbGC], Corollary 2.7, (ii), that the isomorphism ∼ α  y  , y  : Π G i, y → Π G i, y  is graphic. In particular, by allowing y  , y   to vary, it follows immediately from the various definitions involved that Claim 3.17.F holds. This completes the proof of Claim 3.17.F, hence also of assertion (v). Assertion (vi) follows from [NodNon], Theo- rem B; [CbTpII], Theorem A, (i). This completes the proof of Theo- rem 3.17.  COMBINATORIAL ANABELIAN TOPICS III 91 Remark 3.17.1. In the notation of Theorem 3.17, suppose that we are in the situation of Theorem 3.17, (v). Then it follows from Theo- rem 3.17, (v), that C Out F (Π n ) (Out FC (Π n ) M ) ⊆ Out G (Π n ). On the other hand, Out FC (Π n ) M is not, in general, commensurably terminal in Out G (Π n ) [or indeed in Out F (Π n ) or Out FC (Π n )!]. Indeed, suppose, moreover, that we are in the situation of Theorem 3.17, (iii) [so p ∈ Σ], and that the semi-graph of anabelioids G of pro-Σ PSC type log determined by the geometric special fiber of the stable model of X K satisfies the following conditions: • Vert(G)  = Node(G)  = 2. Write Vert(G) = {v 1 , v 2 }, Node(G) = {e 1 , e 2 }. • For each i ∈ {1, 2}, V(e i ) = Vert(G) = {v 1 , v 2 }. • There exists an automorphism of G that induces a nontrivial automorphism of Node(G). Finally, suppose that if we write μ X log for the metric structure on the K log underlying semi-graph of G associated to the stable model of X K [cf. Definition 3.5, (iii)], then μ X log (e 1 )  = μ X log (e 2 ). [Here, we note that one K K log verifies easily that such a smooth log curve X K exists.] Then it follows immediately from the various assumptions imposed on the objects un- der consideration that Out FC (Π 1 ) M is of index 2, hence also normal, in Out G (Π 1 ). In particular, Out FC (Π 1 ) M is not normally terminal, hence, a fortiori, not commensurably terminal, in Out G (Π 1 ). Remark 3.17.2. In the notation of Theorem 3.17, suppose that p ∈ Σ. (i) It follows from Theorem 3.17, (ii-c), that if either († 1 ): n ≥ 4 or n ≥ 3 and r  = 0, then we have equalities Out F (Π n ) M = Out FC (Π n ) M = Out FI (Π n ) = Out F {l}-I (Π n ) = Out FCI (Π n ) = Out FC {l}-I (Π n ) = Out F (Π n ) I = Out F (Π n ) {l}-I = Out FC (Π n ) I = Out FC (Π n ) {l}-I . (ii) In Corollary 2.10, the authors gave what may be regarded as an almost pro-l version of the injectivity portion of [NodNon], Theorem B [i.e., the injectivity of the natural homomorphism Out FC (Π n+1 ) → Out FC (Π n )]. In fact, however, although a de- tailed exposition lies beyond the scope of the present paper [cf. the discussion of (iii) below], it seems quite likely that 92 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI it should be possible to verify an almost pro-l version of the injectivity portion of [CbTpII], Theorem A, (i) [i.e., the injec- tivity of the natural homomorphism Out F (Π n+1 ) → Out F (Π n ) for (r, n)  = (0, 1)]. Such an almost pro-l version would then imply, via a similar argument to the argument applied in the proof of the equalities Out FCI (Π n ) = Out FC (Π n ) I , Out FC {l}-I (Π n ) = Out FC (Π n ) {l}-I [cf. Claim 3.17.A in the proof of Theorem 3.17, (ii)], that if either († 2 ): n ≥ 3 or n ≥ 2 and r  = 0, then the equalities Out FI (Π n ) = Out F (Π n ) I , Out F {l}-I (Π n ) = Out F (Π n ) {l}-I , hence also [cf. Theorem 3.17, (ii); Theorem 3.17, (ii-a)] the nine equalities of the display of (i), hold. (iii) The main reason that the authors did not go to the trouble to verify the nine equalities of the display of (i) under the more general hypotheses [i.e., († 2 )] discussed in (ii) is the following. The main applications of the theory developed in the present paper are the following: (1) the generalization, given in Corollary 3.20 below [cf. also Remark 3.20.1 below], of a result due to Andre [cf. [André], Theorems 7.2.1, 7.2.3] concerning the characterization of local Galois groups in the global Galois image associated to a hyperbolic curve over a number field and (2) the establishment of an appropriate local analogue, sat- isfying various expected properties, of the Grothendieck- Teichmüller group [cf. Remark 3.19.2 below]. The theory surrounding these applications [cf. Theorem 3.18 below] revolves around the theory of the tripod homomorphism developed in [CbTpII], §3. On the other hand, this theory of the tripod homomorphism is only well-behaved [cf. [CbTpII], Definition 3.19] under the more restrictive hypotheses [i.e., († 1 )] discussed in (i). Theorem 3.18 (Metric-admissible outomorphisms and tripods). In the notation of Theorem 3.17, the following hold: (i) Suppose that n ≥ 3. Let Π tpd be a 1-central {1, 2, 3}-tripod of Π n [cf. [CbTpII], Definitions 3.3, (i); 3.7, (ii)]. Then the restriction of the tripod homomorphism associated to Π n T Π tpd : Out FC (Π n ) −→ Out C (Π tpd ) COMBINATORIAL ANABELIAN TOPICS III 93 [cf. [CbTpII], Definition 3.19] to the subgroup Out FC (Π n ) M ⊆ Out FC (Π n ) [cf. Definition 3.7, (iii)] factors through the sub- group Out(Π tpd ) M ⊆ Out C (Π tpd ) [cf. Definition 3.7, (i), (ii); Remark 3.13.1, (i), (ii)], i.e., we have a natural commutative diagram Out FC (Π n ) M −−−→ Out(Π tpd ) M ⏐ ⏐ ⏐ ⏐   Out FC (Π n ) −−−→ Out C (Π tpd ). T Πtpd (ii) Suppose that n ≥ 1, and that (g, r) = (0, 3). Write Out F (Π n ) Δ+ ⊆ Out F (Π n ) for the inverse image via the natural homomorphism Out F (Π n ) → Out(Π 1 ) [cf. [CbTpI], Theorem A, (i)] of Out C (Π 1 ) Δ+ ⊆ Out(Π 1 ) [cf. [CbTpII], Definition 3.4, (i)]; def Out FC (Π n ) Δ+ = Out F (Π n ) Δ+ ∩ Out FC (Π n ) [cf. Remark 3.18.1 below]; def Out F (Π n ) MΔ+ = Out F (Π n ) Δ+ ∩ Out F (Π n ) M ; def Out FC (Π n ) MΔ+ = Out FC (Π n ) Δ+ ∩ Out F (Π n ) M . Then we have equalities Out F (Π n ) Δ+ = Out FC (Π n ) Δ+ , Out F (Π n ) MΔ+ = Out FC (Π n ) MΔ+ . Moreover, the natural homomorphisms Out FC (Π n+1 ) Δ+ −−−→ Out FC (Π n ) Δ+       Out F (Π n+1 ) Δ+ −−−→ Out F (Π n ) Δ+ Out FC (Π n+1 ) MΔ+ −−−→ Out FC (Π n ) MΔ+       Out F (Π n+1 ) MΔ+ −−−→ Out F (Π n ) MΔ+ are bijective. Proof. Assertion (i) follows immediately — in light of the equalities Out FC (Π n ) M = Out FCI (Π n ) , Out(Π tpd ) M = Out I (Π tpd ) ∩ Out C (Π tpd ) 94 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [cf. Theorem 3.17, (i), (ii)] — from the definition of I-admissibility, together with [in the case where Σ = Primes] Corollary 2.13, (iii). Next, we verify assertion (ii). The equalities Out F (Π n ) Δ+ = Out FC (Π n ) Δ+ , Out F (Π n ) MΔ+ = Out FC (Π n ) MΔ+ follow immediately from [CbTpII], Theorem A, (ii), together with the various definitions involved. Next, let us observe that, to verify the bijectivity of the various homomorphisms in question, it suffices to verify the bijectivity of the natural homomorphism Out FC (Π n+1 ) Δ+ −−−→ Out FC (Π n ) Δ+ . On the other hand, this bijectivity follows immediately, in light of the various definitions involved, from [CmbCsp], Corollary 4.2, (i), (ii). This completes the proof of assertion (ii), hence also of Theorem 3.18.  Remark 3.18.1. In the notation of Theorem 3.18, suppose that n ≥ 2. Then in [CmbCsp], Definition 1.11, (ii), a definition was given for the notation “Out FC (Π n ) Δ+ ”, in the case of arbitrary (g, r), that differs somewhat from the definition given for this notation in Theorem 3.18, (ii), when (g, r) = (0, 3). On the other hand, one verifies easily, by applying the theory of [CbTpII], §3, that, when (g, r) = (0, 3), these two definitions are in fact equivalent. Indeed, when n = 2 (respec- tively, n ≥ 3), this follows immediately from [CbTpII], Lemma 3.15, (ii) (respectively, [CbTpII], Theorems 3.16, (v); 3.18, (ii)). Theorem 3.19 (Metric-, graph-admissible outomorphisms and tempered fundamental groups). In the notation of Theorem 3.17, ∧ write K for the p-adic completion of K; ∧ π 1 temp ((X K ) log n × K K ) for the tempered fundamental group [cf. [André], §4, as well as the discussion of Definition 3.1, (ii), of the present paper] of (X K ) log n × K ∧ K ; ∧ def = lim π 1 temp ((X K ) log Π tp n n × K K )/N ←− N ∧ for the Σ-tempered fundamental group of (X K ) log [cf. n × K K [CmbGC], Corollary 2.10, (iii)], i.e., the inverse limit given by allow- ∧ ing N to vary over the open normal subgroups of π 1 temp ((X K ) log n × K K ) such that the quotient by N corresponds to a topological covering [cf. [André], §4.2, as well as the discussion of Definition 3.1, (ii), of the present paper] of some finite log étale Galois covering of ∧ (X K ) log of degree a product of primes ∈ Σ. [Here, we recall n × K K COMBINATORIAL ANABELIAN TOPICS III 95 that, when n = 1, such a “topological covering” corresponds to a “com- binatorial covering”, i.e., a covering determined by a covering of the dual semi-graph of the special fiber of the stable model of some finite ∧ log étale covering of (X K ) log n × K K .] Then the following hold: (i) Let l ∈ Σ be such that l  = p. Then the natural inclusion Out {l}-G (Π n ) → Out(Π n ) [cf. Definition 3.13, (iv)] factors as a composite of homomor- phisms Out {l}-G (Π n ) −→ Out(Π tp n ) −→ Out(Π n ) — where the second arrow is the natural homomorphism [cf. Proposition 3.3, (i)]. In particular, the image of the natural homomorphism Out(Π tp n ) → Out(Π n ) contains the subgroup {l}-G Out (Π n ) ⊆ Out(Π n ), hence also the subgroup Out G (Π n ) ⊆ Out(Π n ) [cf. Definition 3.13, (iv)]. (ii) Write M tp Out FC (Π tp n ) ⊆ Out(Π n ) for the inverse image of Out FC (Π n ) M ⊆ Out(Π n ) [cf. Defi- nition 3.7, (iii)] via the natural homomorphism Out(Π tp n ) → Out(Π n ) [cf. (i)]. Then the resulting natural homomorphism FC M M Out FC (Π tp n ) −→ Out (Π n ) is split surjective, i.e., there exists a homomorphism M Φ : Out FC (Π n ) M −→ Out FC (Π tp n ) such that the composite Φ FC M M Out FC (Π n ) M −→ Out FC (Π tp n ) −→ Out (Π n ) is the identity automorphism of Out FC (Π n ) M . Proof. Assertion (i) follows immediately from Proposition 3.16, (ii). Assertion (ii) follows immediately from assertion (i), together with the fact that Out FC (Π n ) M ⊆ Out {l}-G (Π n ) [cf. Theorem 3.17, (ii)]. This completes the proof of Theorem 3.19.  Remark 3.19.1. In the fourth line of the proof of [André], Proposition 8.6.2, it is asserted that one has an injection alg Aut (Γ alg 0,r+1 ) → Aut (Γ 0,r ) . In the notation of the present series of papers [cf. [CmbCsp], Propo- sition 1.3, (vi), (vii)], this homomorphism corresponds to the natural homomorphism Aut FC (Π n+1 ) cusp −→ Aut FC (Π n ) cusp 96 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI in the case where (g, r, Σ) = (0, 3, Primes), and we observe that Π n for n ≥ 1 corresponds to “Γ alg 0,r for r−3” in the notation of [André], Proposi- alg tion 8.6.2. However, this assertion is false. Indeed, since Γ alg 0,r+1 and Γ 0,r are center-free [cf., e.g., [MzTa], Proposition 2.2, (ii)], it follows that the respective subgroups of inner automorphisms determine compati- alg alg alg ble injections Γ alg 0,r+1 → Aut (Γ 0,r+1 ), Γ 0,r → Aut (Γ 0,r ). On the other alg hand, since the natural surjection Γ alg 0,r+1  Γ 0,r is far from injective, it alg thus follows that the natural homomorphism Aut (Γ alg 0,r+1 ) → Aut (Γ 0,r ) also fails to be injective. In particular, the proof given in [André] of the injectivity of the first displayed homomorphism GT (r+1) −→ GT (r) p p of [André], Proposition 8.6.2, (1) — hence also of • [André], Proposition 8.6.2, (2), • [André], Corollary 8.6.4, • the final portion of [André], Theorem 8.7.1, and • the portion of [André], Corollary 8.7.2, concerning “GT (r) p ” — must be considered incomplete. Moreover, although it is not directly related to the injectivity of the above discussion, we observe in passing [cf. Remark 3.19.4 below for more details] that the discussion of [André], §8, also contains another misleading error. Remark 3.19.2. Recall that, relative to the notation of the present series of papers, the usual Grothendieck-Teichmüller group corresponds to the group def GT = Out F (Π n ) Δ+ = Out FC (Π n ) Δ+ discussed in Theorem 3.18, (ii) [cf. also Remark 3.18.1], in the case where (g, r, Σ) = (0, 3, Primes) [cf. [CmbCsp], Remark 1.11.1]. Thus, from the point of view of the present paper, it seems that one nat- ural candidate for the notion of a local version of the Grothendieck- Teichmüller group is the “metrized Grothendieck-Teichmüller group” def GT M = Out F (Π n ) MΔ+ = Out FC (Π n ) MΔ+ ⊆ GT discussed in Theorem 3.18, (ii), again in the case where (g, r, Σ) = (0, 3, Primes). Here, we recall that each of these groups GT M , GT admits a natural profinite topology, hence, in particular, is compact [cf. Theorem 3.17, (iv)], and, moreover, is independent, up to canonical isomorphism, of the choice of n ≥ 1 [cf. Theorem 3.18, (ii)]. Finally, one verifies immediately from the existence of the natural splitting of the split surjection discussed in Theorem 3.19, (ii) [cf. also the discussion COMBINATORIAL ANABELIAN TOPICS III 97 of the construction of this splitting in the proof of Proposition 3.16, (ii); Remark 3.19.3 below] that, for any positive integer n, one has a natural inclusion GT M → GT (n+3) p [cf. [André], Notation 8.6.1], hence also a natural inclusion GT M → GT p [cf. [André], Definition 8.6.3]. In particular, one obtains a natural outer action of GT M on the “tower” of tempered fundamental groups “(Γ temp 0,r ) r≥4 ” discussed in [André], Corollary 8.6.4, i.e., in the nota- tion of Theorem 3.19 of the present paper, on the system of tempered fundamental groups {Π tp n } n≥1 that is manifestly compatible with the temp tp quotients Π n  Γ 0,n+3 [cf. [André], §8.5]. Remark 3.19.3. The construction of the splitting Φ given in the proof of Theorem 3.19, (ii), appears, at first glance, to depend on the choice of the prime l, as well as on the ordering of the n factors of the con- figuration spaces that give rise to Π n , Π tp n . In fact, however, it is not difficult to verify — by • observing that symmetries [e.g., that arise from permutations of the n factors] of finite étale coverings of the various configu- ration spaces over fields that appear always extend to symme- tries of the corresponding stable polycurves [cf. the discussion of Remark 3.10.1, (i), (e); [ExtFam], Theorem A]; • applying the functoriality of the various constructions involved [cf. the discussion of “functorial bijections” in the proof of Proposition 3.6] to relate the “decomposition groups” of the various strata that appear in the proof of Proposition 3.16, (ii); and • observing that these strata may be described in terms of “jumps” in the rank of the group-characteristic sheaf [cf. [MzTa], Def- inition 5.1, (i)] associated to the log structure of the stable polycurves that appear [cf. the discussion of Remark 3.10.1, (i), (e)], hence are independent of the ordering of the n factors of the configuration spaces that appear — that Φ is independent of the choice of l, as well as of the ordering of the n factors of the configuration spaces that give rise to Π n , Π tp n . Remark 3.19.4. In passing, we observe, relative to the discussion of [André], §8, that the second isomorphism Out  π 1 top ((P 1 C ) an \ {x 1 , . . . , x r }) ∼ = Ker[OutF r−1 → GL r−1 (Z)] 98 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI of the final display of [André], §8.2, is false as stated and should be replaced by an inclusion arrow “→”. Indeed, let us first observe that the first isomorphism of the final display of [André], §8.2, is correct as stated — and indeed is a special case of the well-known theorem of Dehn-Nielsen-Baer — if one interprets the phrase “local monodromies” in the definition of “Out  ” as referring to generators γ i of the inertia def groups at the points x i . Write Π = π 1 top ((P 1 C ) an \ {x 1 , . . . , x r }). Then the falsity of the second isomorphism — i.e., the non-surjectivity of ∼ the natural inclusion “→” induced by an isomorphism Π → F r−1 — may be verified as follows. First, we observe that if one assumes the surjectivity of this natural inclusion, then it follows that the subgroup Out  (Π) ⊆ Out(Π) is normal. Thus, it suffices to obtain a contradiction under the assumption that the subgroup Out  (Π) ⊆ Out(Π) is normal. Next, let us observe that the discrete free group Π on r − 1 generators is generated by γ 1 , . . . , γ r−1 . In particular, for any element δ ∈ Π that appears as one of a collection of r − 1 generators of Π, there exists an element φ ∈ Out(Π) such that φ(δ) = γ 1 . Thus, since Out  (Π) ⊆ Out(Π) is normal, it follows that any element of Out  (Π) preserves the def def conjugacy class of δ. Write δ 1 = γ 1 ·γ 2 ; for i = 2, . . . , r−3, δ i = δ i−1 ·γ i . Then, by taking “δ” to be δ 1 , . . . , δ r−3 , we conclude that any element of Out  (Π) preserves the conjugacy classes of each of δ 1 , . . . , δ r−3 . On the other hand, one verifies immediately that, for a standard choice of generators γ 1 , . . . , γ r−1 , the elements δ 1 , . . . , δ r−3 may be regarded as generators of the nodal inertia groups associated to the nodes that appear in a totally degenerate pointed stable curve [over the field of complex numbers] that arises as a degeneration of the pointed stable curve corresponding to the given Riemann surface (P 1 C ) an . Thus, it follows from [CbTpIV], Corollary 2.19, (i); [CmbGC], Theorem 1.6, (ii); [CmbGC], Proposition 1.3, that any element of Out  (Π) is graphic, i.e., in particular, preserves the conjugacy classes of the verticial and nodal subgroups of Π that arise from this totally degenerate structure. On the other hand, in light of [CbTpIV], Corollary 2.21, (iii); [CbTpIV], Theorem 2.24, (ii), this implies that Out  (Π) is an extension of a finite group by an abelian group, i.e., in contradiction to the fact that Out  (Π) admits a surjection to a [highly nonabelian!] discrete free group of rank ≥ 2. Remark 3.19.5. Finally, in passing, we note the following conse- quence, in the context of [Tsjm], of the theory developed in the present paper. Suppose, in the notation of Theorem 3.18, (i), that n ≥ 4 or r > 0. Then the homomorphism Out FC (Π n ) M → Out(Π tpd ) M of COMBINATORIAL ANABELIAN TOPICS III 99 Theorem 3.18, (i), determines [cf. [CbTpII], Definition 3.19] a homo- morphism Out F (Π n ) M → Out FC (Π tpd ) MΔ+ = GT M [cf. the notation of Definition 3.7, (iii); Theorem 3.18, (ii); Remark 3.19.2]. In particular, by composing this last homomorphism with the restric- tion to GT M of the homomorphism of [Tsjm], Corollary B [cf. also [Tsjm], Remark 2.1.2], we obtain a natural composite homomorphism Out F (Π n ) M → GT M → G Q p , whose restriction, via the natural homomorphism I K → Out F (Π n ) M [cf. the homomorphism “ρ n ” of Theorem 3.17], to I K coincides with the natural outer homomorphism I K → G Q p that arises from the natural inclusion of topological fields Q p → K. Corollary 3.20 (Characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve). Let F be a number field, i.e., a finite extension of the field of rational numbers; p a nonarchimedean prime of F ; F p an algebraic closure of the p-adic completion F p of F ; F ⊆ F p the algebraic closure of F in ∧ F p ; X F log a smooth log curve over F . Write F p for the completion def def def of F p ; G p = Gal(F p /F p ) ⊆ G F = Gal(F /F ); X F log = X F log × F F ; π 1 (X F log ) for the log fundamental group of X F log [which, in the following, we ∧ identify with the log fundamental groups of X F log × F F p , X F log × F F p — cf. the definition of F !]; ∧ π 1 temp (X F log × F F p ) ∧ for the tempered fundamental group of X F log × F F p [cf. [André], §4]; ρ X log : G F −→ Out(π 1 (X F log )) F for the natural outer Galois action associated to X F log ; ∧ : G p −→ Out(π 1 temp (X F log × F F p )) ρ temp X log ,p F for the natural outer Galois action associated to X F log × F F p [cf. [André], Proposition 5.1.1]; ∧ Out(π 1 (X F log )) M ⊆ ( Out(π 1 temp (X F log × F F p )) ⊆ ) Out(π 1 (X F log )) for the subgroup of M-admissible outomorphisms of π 1 (X F log ) [cf. Def- inition 3.7, (i), (ii); Proposition 3.6, (i)]. Then the following hold: 100 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (i) The outer Galois action ρ temp factors through the subgroup X log ,p F ∧ Out(π 1 (X F log )) M ⊆ Out(π 1 temp (X F log × F F p )). (ii) We have a natural commutative diagram G p −−−→ Out(π 1 (X F log )) M ⏐ ⏐ ⏐ ⏐   ρ X log F → Out(π 1 (X F log )) G F −−− — where the vertical arrows are the natural inclusions, the upper horizontal arrow is the homomorphism arising from the factorization of (i), and all arrows are injective. (iii) The diagram of (ii) is cartesian, i.e., if we regard the various groups involved as subgroups of Out(π 1 (X F log )), then we have an equality G p = G F ∩ Out(π 1 (X F log )) M . Proof. Assertion (i) follows immediately from the various definitions involved. Assertion (ii) follows immediately from the injectivity of the lower horizontal arrow ρ X log [cf. [NodNon], Theorem C], together with F the various definitions involved. Finally, we verify assertion (iii). First, let us observe that if the smooth log curve “X F log ” is the smooth log curve associated to P 1 F \ {0, 1, ∞}, then assertion (iii) follows immedi- ately from [André], Theorem 7.2.1. Write (X F ) log 3 for the 3-rd log con- log figuration space of X F . Then it follows immediately from [NodNon], Theorem B, that the group Out FC (π 1 ((X F ) log 3 )) of FC-admissible outo- log morphisms of the log fundamental group π 1 ((X F ) log 3 ) of (X F ) 3 [which, in the following, we identify with the log fundamental groups of (X F ) log 3 × F ∧ log F p , (X F ) 3 × F F p — cf. the definition of F !] may be regarded as a closed subgroup of Out(π 1 (X F log )). Moreover, it follows imme- diately from the various definitions involved that the respective images Im(ρ X log ), Im(ρ temp ) of the natural outer Galois actions ρ X log , ρ temp X log ,p X log ,p F F F F associated to X F log , X F log × F F p are contained in this closed subgroup log Out FC (π 1 ((X F ) log 3 )) ⊆ Out(π 1 (X F )). Thus, to verify assertion (iii), one verifies easily that it suffices to verify the equality M Im(ρ temp ) = Im(ρ X log ) ∩ Out FC (π 1 ((X F ) log 3 )) X log ,p F F [cf. Definition 3.7, (iii)]. On the other hand, since the “ρ X log ” that F occurs in the case where we take “X F log ” to be the smooth log curve associated to P 1 F \ {0, 1, ∞} is injective [cf. assertion (ii)], this equality COMBINATORIAL ANABELIAN TOPICS III 101 follows immediately — by considering the images of the subgroups M Im(ρ temp ) ⊆ Im(ρ X log ) ∩ Out FC (π 1 ((X F ) log 3 )) X log ,p F F M of Out FC (π 1 ((X F ) log via the tripod homomorphism associated to 3 )) log FC Out (π 1 ((X F ) 3 )) [cf. [CbTpII], Definition 3.19] — from Theorem 3.18, (i), together with assertion (iii) in the case where we take “X F log ” to be the smooth log curve associated to P 1 F \ {0, 1, ∞} [which was ver- ified above]. This completes the proof of assertion (iii), hence also of Corollary 3.20.  Remark 3.20.1. Corollary 3.20, (iii), may be regarded as a gen- eralization of [André], Theorems 7.2.1, 7.2.3, obtained at the cost of replacing, in effect, Out(π 1 (X F log )) G by the possibly smaller group Out(π 1 (X F log )) M ⊆ Out(π 1 (X F log )). Here, we note that unlike the sub- groups G p ⊆ G F [cf., e.g., [AbsHyp], Theorem 1.1.1, (i)] and Out(π 1 temp ∧ ∼ (X F log × F F p )) → Out(π 1 (X F log )) G ⊆ Out(π 1 (X F log )) [cf. Definition 3.7, (i); Proposition 3.6, (i); Remark 3.13.1, (i); Theorem 3.17, (v)], which are commensurably terminal, the subgroup Out(−) M ⊆ Out(−) fails, in general [at least in the pro-l case], even to be normally terminal [cf. Remark 3.17.1]. Remark 3.20.2. Let us recall that, in the proof of [NodNon], Theorem C, the authors applied • the injectivity portion of the theory of combinatorial cuspidal- ization, together with • the injectivity of the outer Galois representation associated to a tripod, to prove • the injectivity of the outer Galois representation associated to an arbitrary hyperbolic curve. On the other hand, in the proof of Corollary 3.20, the authors applied • the [almost pro-l] injectivity portion of the theory of combinato- rial cuspidalization [in the form of Theorem 3.18, (i)], together with • the characterization of the local Galois groups in the global Galois image for tripods, to prove • an analogous characterization of the local Galois groups in the global Galois image for arbitrary hyperbolic curves. 102 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI The formal similarity of these two proofs suggests that it is perhaps natural to think of the injectivity portion of the theory of combina- torial cuspidalization as a sort of tool for reducing certain problems concerning arbitrary hyperbolic curves to the case of tripods. Remark 3.20.3. By comparison to André’s original characterization of the local Galois groups in the global Galois image [cf. 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